Problem: On the xy-plane, Snuke is going to travel from the point (x_s, y_s) to the point (x_t, y_t). He can move in arbitrary directions with speed 1. Here, we will consider him as a point without size.

There are N circular barriers deployed on the plane. The center and the radius of the i-th barrier are (x_i, y_i) and r_i, respectively. The barriers may overlap or contain each other.

A point on the plane is exposed to cosmic rays if the point is not within any of the barriers.

Snuke wants to avoid exposure to cosmic rays as much as possible during the travel. Find the minimum possible duration of time he is exposed to cosmic rays during the travel.

Constraints

* All input values are integers.
* -10^9 ≤ x_s, y_s, x_t, y_t ≤ 10^9
* (x_s, y_s) ≠ (x_t, y_t)
* 1≤N≤1,000
* -10^9 ≤ x_i, y_i ≤ 10^9
* 1 ≤ r_i ≤ 10^9

Input

The input is given from Standard Input in the following format:


x_s y_s x_t y_t
N
x_1 y_1 r_1
x_2 y_2 r_2
:
x_N y_N r_N


Output

Print the minimum possible duration of time Snuke is exposed to cosmic rays during the travel. The output is considered correct if the absolute or relative error is at most 10^{-9}.

Examples

Input

-2 -2 2 2
1
0 0 1


Output

3.6568542495


Input

-2 0 2 0
2
-1 0 2
1 0 2


Output

0.0000000000


Input

4 -2 -2 4
3
0 0 2
4 0 1
0 4 1


Output

4.0000000000

Solution (in JAVA):

Solution:
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.lang.reflect.Array;
import java.math.BigDecimal;
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.Collection;
import java.util.Comparator;
import java.util.HashMap;
import java.util.Map.Entry;
import java.util.NoSuchElementException;
import java.util.Set;
import java.util.TreeMap;
import java.util.TreeSet;

public class Main {
	private static void solve() {
		long sx = nl();
		long sy = nl();
		long ex = nl();
		long ey = nl();
		int n = ni() + 2;
		long[][] xyr = nls2(n - 2, 3);
		double[][] dist = new double[n][n];
		for (int i = 0; i < n; i++) {
			long x1 = i == n - 2 ? sx : i == n - 1 ? ex : xyr[i][0];
			long y1 = i == n - 2 ? sy : i == n - 1 ? ey : xyr[i][1];
			long r1 = i >= n - 2 ? 0 : xyr[i][2];
			for (int j = 0; j < i; j++) {
				long x2 = j == n - 2 ? sx : j == n - 1 ? ex : xyr[j][0];
				long y2 = j == n - 2 ? sy : j == n - 1 ? ey : xyr[j][1];
				long r2 = j >= n - 2 ? 0 : xyr[j][2];

				long dx = x1 - x2;
				long dy = y1 - y2;
				double d = Math.sqrt(dx * dx + dy * dy);
				double gap = d - r1 - r2;
				if (gap < 0)
					gap = 0;
				dist[i][j] = gap;
				dist[j][i] = gap;
			}
		}
		double[][] res = Graph.dijkstraD(n, n - 2, i -> n - 1, (i, f) -> {
			for (int j = 0; j < n; j++) {
				if (i == j)
					continue;
				f.f(j, dist[i][j]);
			}
		});
		out(res[n - 1][0]);
	}

	static int min(int a, int b) {
		return a < b ? a : b;
	}

	static long min(long a, long b) {
		return a < b ? a : b;
	}

	static <A extends Comparable<? super A>> A min(A a, A b) {
		return a.compareTo(b) < 0 ? a : b;
	}

	static int max(int a, int b) {
		return a > b ? a : b;
	}

	static long max(long a, long b) {
		return a > b ? a : b;
	}

	static <A extends Comparable<? super A>> A max(A a, A b) {
		return a.compareTo(b) > 0 ? a : b;
	}

	static int clamp(int a, int min, int max) {
		return a < min ? min : a > max ? max : a;
	}

	static long clamp(long a, long min, long max) {
		return a < min ? min : a > max ? max : a;
	}

	static int abs(int a) {
		return a < 0 ? -a : a;
	}

	static long abs(long a) {
		return a < 0 ? -a : a;
	}

	static <A extends Comparable<? super A>> A clamp(A a, A min, A max) {
		return a.compareTo(min) < 0 ? min : a.compareTo(max) > 0 ? max : a;
	}

	static void out(String val) {
		IO.out(val);
	}

	static void out(Object val) {
		IO.out(String.valueOf(val));
	}

	static void out(int val) {
		IO.out(String.valueOf(val));
	}

	static void out(long val) {
		IO.out(String.valueOf(val));
	}

	static void out(char val) {
		IO.out(String.valueOf(val));
	}

	static void out(double val) {
		IO.out(Double.isFinite(val) ? BigDecimal.valueOf(val).toPlainString() : String.valueOf(val));
	}

	static void out(boolean val) {
		IO.out(val ? "true" : "false");
	}

	static void outn(String val) {
		IO.outn(val);
	}

	static void outn(Object val) {
		IO.outn(String.valueOf(val));
	}

	static void outn(int val) {
		IO.outn(String.valueOf(val));
	}

	static void outn(long val) {
		IO.outn(String.valueOf(val));
	}

	static void outn(char val) {
		IO.outn(String.valueOf(val));
	}

	static void outn(double val) {
		IO.outn(Double.isFinite(val) ? BigDecimal.valueOf(val).toPlainString() : String.valueOf(val));
	}

	static void outn(boolean val) {
		IO.outn(val ? "true" : "false");
	}

	static void kil(String val) {
		IO.out(val);
		IO.flush();
		System.exit(0);
	}

	static void kil(Object val) {
		IO.out(String.valueOf(val));
		IO.flush();
		System.exit(0);
	}

	static void kil(int val) {
		IO.out(String.valueOf(val));
		IO.flush();
		System.exit(0);
	}

	static void kil(long val) {
		IO.out(String.valueOf(val));
		IO.flush();
		System.exit(0);
	}

	static void kil(char val) {
		IO.out(String.valueOf(val));
		IO.flush();
		System.exit(0);
	}

	static void kil(double val) {
		IO.out(Double.isFinite(val) ? BigDecimal.valueOf(val).toPlainString() : String.valueOf(val));
		IO.flush();
		System.exit(0);
	}

	static void kil(boolean val) {
		IO.out(val ? "true" : "false");
		IO.flush();
		System.exit(0);
	}

	static String ns() {
		return IO.next();
	}

	static int ni() {
		return IO.nextInt();
	}

	static long nl() {
		return IO.nextLong();
	}

	static double nd() {
		return IO.nextDouble();
	}

	static char nc() {
		return IO.nextChar();
	}

	static String[] nss(int n) {
		String[] as = new String[n];
		for (int i = 0; i < n; i++) {
			as[i] = IO.next();
		}
		return as;
	}

	static int[] nis(int n) {
		int[] as = new int[n];
		for (int i = 0; i < n; i++) {
			as[i] = IO.nextInt();
		}
		return as;
	}

	static long[] nls(int n) {
		long[] as = new long[n];
		for (int i = 0; i < n; i++) {
			as[i] = IO.nextLong();
		}
		return as;
	}

	static double[] nds(int n) {
		double[] as = new double[n];
		for (int i = 0; i < n; i++) {
			as[i] = IO.nextDouble();
		}
		return as;
	}

	static char[] ncs(int n) {
		char[] as = new char[n];
		for (int i = 0; i < n; i++) {
			as[i] = IO.nextChar();
		}
		return as;
	}

	static String[][] nss2(int n, int m) {
		String[][] as = new String[n][m];
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < m; j++) {
				as[i][j] = IO.next();
			}
		}
		return as;
	}

	static int[][] nis2(int n, int m) {
		int[][] as = new int[n][m];
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < m; j++) {
				as[i][j] = IO.nextInt();
			}
		}
		return as;
	}

	static long[][] nls2(int n, int m) {
		long[][] as = new long[n][m];
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < m; j++) {
				as[i][j] = IO.nextLong();
			}
		}
		return as;
	}

	static double[][] nds2(int n, int m) {
		double[][] as = new double[n][m];
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < m; j++) {
				as[i][j] = IO.nextDouble();
			}
		}
		return as;
	}

	static char[][] ncs2(int n, int m) {
		char[][] as = new char[n][m];
		for (int i = 0; i < n; i++) {
			for (int j = 0; j < m; j++) {
				as[i][j] = IO.nextChar();
			}
		}
		return as;
	}

	static int parseInt(String val) {
		return Integer.parseInt(val);
	}

	static long parseLong(String val) {
		return Long.parseLong(val);
	}

	public static void main(String[] args) {
		try {
			solve();
			IO.flush();
		} catch (NumberFormatException e) {
			e.printStackTrace();
		} catch (OutOfMemoryError e) {
			e.printStackTrace(); // this will be detected as RE
		}
	}
}

final class IO {
	private static final InputStream in = System.in;
	private static final PrintWriter out = new PrintWriter(System.out, false);
	private static final byte[] buffer = new byte[1024];
	private static int ptr = 0;
	private static int len = 0;

	private static boolean hasNextByte() {
		if (ptr < len)
			return true;
		ptr = 0;
		try {
			len = in.read(buffer);
		} catch (IOException e) {
			e.printStackTrace();
		}
		return len > 0;
	}

	private static int readByte() {
		if (hasNextByte())
			return buffer[ptr++];
		else
			return -1;
	}

	static boolean hasNext() {
		byte c;
		while (hasNextByte() && ((c = buffer[ptr]) < '!' || c > '~'))
			ptr++;
		return hasNextByte();
	}

	static String next() {
		if (!hasNext())
			throw new NoSuchElementException();
		StringBuilder sb = new StringBuilder();
		int b = readByte();
		while (b >= '!' && b <= '~') {
			sb.append((char) b);
			b = readByte();
		}
		return sb.toString();
	}

	static char nextChar() {
		if (!hasNext())
			throw new NoSuchElementException();
		return (char) readByte();
	}

	static long nextLong() {
		if (!hasNext())
			throw new NoSuchElementException();
		long n = 0;
		int sign = 1;
		int b = readByte();
		if (b == '-') {
			sign = -1;
			b = readByte();
		}
		if (b < '0' || '9' < b)
			throw new NumberFormatException();
		while (true) {
			if ('0' <= b && b <= '9')
				n = n * 10 + b - '0';
			else if (b == -1 || b < '!' || b > '~')
				return n * sign;
			else
				throw new NumberFormatException();
			b = readByte();
		}
	}

	static int nextInt() {
		if (!hasNext())
			throw new NoSuchElementException();
		int n = 0;
		int sign = 1;
		int b = readByte();
		if (b == '-') {
			sign = -1;
			b = readByte();
		}
		if (b < '0' || '9' < b)
			throw new NumberFormatException();
		while (true) {
			if ('0' <= b && b <= '9')
				n = n * 10 + b - '0';
			else if (b == -1 || b < '!' || b > '~')
				return n * sign;
			else
				throw new NumberFormatException();
			b = readByte();
		}
	}

	static double nextDouble() {
		return Double.parseDouble(next());
	}

	static void out(String val) {
		out.println(val);
	}

	static void outn(String val) {
		out.print(val);
	}

	static void flush() {
		out.flush();
	}
}

// TODO: paste library here

class UF<A> { // Union Find Tree
	private int[] parents;
	private int[] ranks;
	private ArrayList<A> roots;
	private F.XXX<A, A, A> merger;

	UF(int n, F.IX<A> maker, F.XXX<A, A, A> merger) {
		this.merger = merger;
		parents = new int[n];
		ranks = new int[n];
		roots = U.make(n, maker);

		for (int i = 0; i < n; i++) {
			parents[i] = i;
		}
	}

	int index(int i) {
		int p = parents[i];
		if (i == p) {
			return i;
		}
		return parents[i] = index(p);
	}

	boolean same(int i, int j) {
		return index(i) == index(j);
	}

	A root(int i) {
		return roots.get(index(i));
	}

	void unite(int i, int j) {
		i = index(i);
		j = index(j);
		if (i == j)
			return;
		A r = merger.f(root(i), root(j));
		if (ranks[i] < ranks[j]) {
			i ^= j;
			j ^= i;
			i ^= j;
		}
		parents[j] = i;
		roots.set(i, r);
		if (ranks[i] == ranks[j])
			ranks[i]++;
	}
}

class MS<A> {
	private TreeMap<A, Long> tm;

	MS() {
		tm = new TreeMap<A, Long>();
	}

	MS(Comparator<A> comp) {
		tm = new TreeMap<A, Long>(comp);
	}

	void add(A a, long num) {
		if (tm.containsKey(a))
			tm.put(a, tm.get(a) + num);
		else
			tm.put(a, num);
	}

	void add(A a) {
		add(a, 1);
	}

	long remove(A a, long num) {
		if (tm.containsKey(a)) {
			long n = tm.get(a);
			if (n <= num) {
				tm.remove(a);
				return n;
			}
			tm.put(a, n - num);
			return num;
		}
		return 0;
	}

	boolean remove(A a) {
		return remove(a, 1) == 1;
	}

	long removeAll(A a) {
		return remove(a, Long.MAX_VALUE);
	}

	long num(A a) {
		return tm.containsKey(a) ? tm.get(a) : 0;
	}

	int size() {
		return tm.size();
	}

	A floor(A a) {
		return tm.floorKey(a);
	}

	UP<A, Long> floorNum(A a) {
		Entry<A, Long> e = tm.floorEntry(a);
		return e == null ? null : UP.make(e.getKey(), e.getValue());
	}

	A ceil(A a) {
		return tm.ceilingKey(a);
	}

	UP<A, Long> ceilNum(A a) {
		Entry<A, Long> e = tm.ceilingEntry(a);
		return e == null ? null : UP.make(e.getKey(), e.getValue());
	}

	void clear() {
		tm.clear();
	}

	Set<A> keySet() {
		return tm.keySet();
	}

	Collection<Long> valueSet() {
		return tm.values();
	}

	Set<Entry<A, Long>> entrySet() {
		return tm.entrySet();
	}

	public String toString() {
		return tm.toString();
	}
}

class WUF<A, W> { // Weighted Union Find Tree
	private int[] parents;
	private int[] ranks;
	private ArrayList<A> roots;
	private ArrayList<W> weights;
	private F.XXX<A, A, A> merger;
	private F.XXX<W, W, W> add;
	private F.XX<W, W> inv;
	private W e;

	WUF(int n, F.IX<A> maker, F.XXX<A, A, A> merger, F.XXX<W, W, W> add, F.XX<W, W> inv, W e) {
		this.merger = merger;
		this.add = add;
		this.inv = inv;
		this.e = e;
		parents = new int[n];
		ranks = new int[n];
		roots = U.make(n, maker);
		weights = U.make(n, i -> e);

		for (int i = 0; i < n; i++) {
			parents[i] = i;
		}
	}

	int index(int i) {
		int p = parents[i];
		if (i == p) {
			return i;
		}
		int r = index(p);
		weights.set(i, add.f(weights.get(i), weights.get(p)));
		return parents[i] = r;
	}

	boolean same(int i, int j) {
		return index(i) == index(j);
	}

	A root(int i) {
		return roots.get(index(i));
	}

	W weight(int i) {
		index(i);
		return weights.get(i);
	}

	W diff(int i, int j) {
		return sub(weight(j), weight(i));
	}

	void unite(int i, int j, W w) {
		w = add(w, weight(i));
		w = sub(w, weight(j));
		i = index(i);
		j = index(j);
		if (i == j)
			return;
		A r = merger.f(root(i), root(j));
		if (ranks[i] < ranks[j]) {
			i ^= j;
			j ^= i;
			i ^= j;
			w = inv.f(w);
		}
		parents[j] = i;
		roots.set(i, r);
		if (ranks[i] == ranks[j])
			ranks[i]++;
		weights.set(j, w);
	}

	private W add(W a, W b) {
		return a == e ? b : b == e ? a : add.f(a, b);
	}

	private W sub(W a, W b) {
		return b == e ? a : add(a, inv.f(b));
	}
}

class PQ<A> { // Priority Queue
	static class E { // Entry
		private Object a;
		private int index;

		private E(Object a, int index) {
			this.a = a;
			this.index = index;
		}
	}

	private int n;
	private E[] as;
	private Comparator<A> comp;

	PQ(Comparator<A> comp) {
		this.comp = comp;
		as = new E[64];
		n = 1;
	}

	E add(A a) {
		E res = addEntry(a);
		fixUp(n - 1);
		return res;
	}

	A pop() {
		int max = n - 1;
		if (max == 0)
			return null;
		if (max == 1) {
			return removeLast();
		}

		E tmp = as[1];
		as[1] = as[max];
		as[max] = tmp;
		as[1].index = 1;
		as[max].index = max;

		A res = removeLast();
		fixDown(1);
		return res;
	}

	@SuppressWarnings("unchecked")
	A get(E e) {
		return (A) e.a;
	}

	void remove(E e) {
		int k = ((E) e).index;
		int max = n - 1;
		if (k == max) {
			removeLast();
			return;
		}

		E tmp = as[k];
		as[k] = as[max];
		as[max] = tmp;
		as[k].index = k;
		as[max].index = max;

		removeLast();
		if (!fixDown(k))
			fixUp(k);
	}

	void update(E e, A a) {
		int k = e.index;
		as[k].a = a;
		if (!fixDown(k))
			fixUp(k);
	}

	boolean isEmpty() {
		return n == 1;
	}

	@SuppressWarnings("unchecked")
	private A removeLast() {
		n--;
		E res = as[n];
		res.index = -1;
		as[n] = null;
		return (A) res.a;
	}

	@SuppressWarnings("unchecked")
	private boolean fixUp(int k) {
		boolean res = false;
		while (k > 1) {
			int nk = k >> 1;
			if (comp.compare((A) as[k].a, (A) as[nk].a) < 0) {
				E tmp = as[k];
				as[k] = as[nk];
				as[nk] = tmp;
				as[k].index = k;
				as[nk].index = nk;
				k = nk;
				res = true;
				continue;
			}
			break;
		}
		return res;
	}

	@SuppressWarnings("unchecked")
	private boolean fixDown(int k) {
		boolean res = false;
		while (k << 1 < n) {
			A a = (A) as[k].a;
			A l = (A) as[k << 1].a;
			if (k << 1 == n - 1) {
				if (comp.compare(a, l) <= 0)
					break;
				res = true;
				E tmp = as[k];
				as[k] = as[k << 1];
				as[k << 1] = tmp;
				as[k].index = k;
				as[k << 1].index = k << 1;
				break;
			}
			A r = (A) as[k << 1 | 1].a;
			if (comp.compare(a, l) <= 0 && comp.compare(a, r) <= 0)
				break;
			res = true;
			if (comp.compare(l, r) < 0) {
				int nk = k << 1;
				E tmp = as[k];
				as[k] = as[nk];
				as[nk] = tmp;
				as[k].index = k;
				as[nk].index = nk;
				k = nk;
			} else {
				int nk = k << 1 | 1;
				E tmp = as[k];
				as[k] = as[nk];
				as[nk] = tmp;
				as[k].index = k;
				as[nk].index = nk;
				k = nk;
			}
		}
		return res;
	}

	private E addEntry(A a) {
		if (n == as.length)
			as = U.doubleSize(as);
		return as[n] = new E(a, n++);
	}

	@SuppressWarnings("unchecked")
	public String toString() {
		if (n == 1)
			return "[]";
		String s = null;
		for (int i = 1; i < n; i++) {
			A a = (A) as[i].a;
			if (s == null)
				s = "[" + a;
			else
				s += ", " + a;
		}
		return s + "]";
	}
}

class Graph {
	/**
	 * !!多重辺があるとバグる!!
	 * !!多重辺があるとバグる!!
	 * !!多重辺があるとバグる!!
	 * 
	 * @param forEachEdge
	 *            (index:int, (edgeTo:int, edgeCost:long) -> void) -> void
	 * @return
	 *         [0] = dist, [1] = prev
	 */
	static long[][] dijkstraL(int n, int from, F.II numEdges, F.IXV<F.ILV> forEachEdge) {
		long inf = Long.MAX_VALUE;
		long[][] res = new long[n][2];
		for (int i = 0; i < n; i++) {
			res[i][0] = i == from ? 0 : inf;
			res[i][1] = -1;
		}
		PQ<long[]> q = new PQ<long[]>((a, b) -> Long.compare(a[1], b[1]));

		PQ.E[] qes = new PQ.E[n];
		ArrayList<long[]> ups = new ArrayList<long[]>(n);
		for (int i = 0; i < n; i++) {
			ups.add(new long[] { i, inf });
		}

		boolean[] fixed = new boolean[n];
		ups.get(from)[1] = 0;
		qes[from] = q.add(ups.get(from));

		while (!q.isEmpty()) {
			long[] up = q.pop();
			int i = (int) up[0];
			if (fixed[i])
				continue;
			fixed[i] = true;
			long dist = up[1];
			if (dist == inf)
				break;
			forEachEdge.f(i, (j, weight) -> {
				if (fixed[j])
					return;
				long[] resj = res[j];
				long newDist = dist + weight;
				if (Long.compare(resj[0], newDist) > 0) {
					resj[0] = newDist;
					resj[1] = i;
					q.add(new long[] { j, newDist });
				}
			});
		}
		return res;
	}

	/**
	 * !!多重辺があるとバグる!!
	 * !!多重辺があるとバグる!!
	 * !!多重辺があるとバグる!!
	 * 
	 * @param forEachEdge
	 *            (index:int, (edgeTo:int, edgeCost:long) -> void) -> void
	 * @return
	 *         [0] = dist, [1] = prev
	 */
	static double[][] dijkstraD(int n, int from, F.II numEdges, F.IXV<F.IDV> forEachEdge) {
		double inf = Double.POSITIVE_INFINITY;
		double[][] res = new double[n][2];
		for (int i = 0; i < n; i++) {
			res[i][0] = i == from ? 0 : inf;
			res[i][1] = -1;
		}
		PQ<double[]> q = new PQ<double[]>((a, b) -> Double.compare(a[1], b[1]));

		PQ.E[] qes = new PQ.E[n];
		ArrayList<double[]> ups = new ArrayList<double[]>(n);
		for (int i = 0; i < n; i++) {
			ups.add(new double[] { i, inf });
		}

		boolean[] fixed = new boolean[n];
		ups.get(from)[1] = 0;
		qes[from] = q.add(ups.get(from));

		while (!q.isEmpty()) {
			double[] up = q.pop();
			int i = (int) up[0];
			if (fixed[i])
				continue;
			fixed[i] = true;
			double dist = up[1];
			if (dist == inf)
				break;
			forEachEdge.f(i, (j, weight) -> {
				if (fixed[j])
					return;
				double[] resj = res[j];
				double newDist = dist + weight;
				if (Double.compare(resj[0], newDist) > 0) {
					resj[0] = newDist;
					resj[1] = i;
					q.add(new double[] { j, newDist });
				}
			});
		}
		return res;
	}

}

class UP<A, B> { // Unordered Pair
	A a;
	B b;

	UP(A a, B b) {
		this.a = a;
		this.b = b;
	}

	static <A, B> UP<A, B> make(A a, B b) {
		return new UP<A, B>(a, b);
	}

	public boolean equals(Object o) {
		if (o == this)
			return true;
		if (!(o instanceof UP))
			return false;
		UP<?, ?> p = (UP<?, ?>) o;
		boolean aok = a == null ? p.a == null : a.equals(p.a);
		boolean bok = b == null ? p.b == null : b.equals(p.b);
		return aok && bok;
	}

	public String toString() {
		return "(" + a.toString() + ", " + b.toString() + ")";
	}
}

class P<A extends Comparable<? super A>, B extends Comparable<? super B>> extends UP<A, B>
		implements Comparable<P<A, B>> { // Pair
	P(A a, B b) {
		super(a, b);
	}

	static <A extends Comparable<? super A>, B extends Comparable<? super B>> P<A, B> make(A a, B b) {
		return new P<A, B>(a, b);
	}

	public int compareTo(P<A, B> o) {
		int sa = a.compareTo(o.a);
		int sb = b.compareTo(o.b);
		return sa != 0 ? sa : sb;
	}
}

class PI implements Comparable<PI> { // Pair int
	int a;
	int b;

	PI(int a, int b) {
		this.a = a;
		this.b = b;
	}

	static PI make(int a, int b) {
		return new PI(a, b);
	}

	public boolean equals(Object o) {
		if (o == this)
			return true;
		if (!(o instanceof PI))
			return false;
		PI p = (PI) o;
		return a == p.a && b == p.b;
	}

	public int compareTo(PI o) {
		int sa = a - o.a;
		int sb = b - o.b;
		return sa > 0 ? 1 : sa < 0 ? -1 : sb > 0 ? 1 : sb < 0 ? -1 : 0;
	}

	public String toString() {
		return "(" + a + ", " + b + ")";
	}
}

class PL implements Comparable<PL> { // Pair long
	long a;
	long b;

	PL(long a, long b) {
		this.a = a;
		this.b = b;
	}

	static PL make(long a, long b) {
		return new PL(a, b);
	}

	public boolean equals(Object o) {
		if (o == this)
			return true;
		if (!(o instanceof PL))
			return false;
		PL p = (PL) o;
		return a == p.a && b == p.b;
	}

	public int compareTo(PL o) {
		long sa = a - o.a;
		long sb = b - o.b;
		return sa > 0 ? 1 : sa < 0 ? -1 : sb > 0 ? 1 : sb < 0 ? -1 : 0;
	}

	public String toString() {
		return "(" + a + ", " + b + ")";
	}
}

class UT<A, B, C> { // Unordered Tuple
	A a;
	B b;
	C c;

	UT(A a, B b, C c) {
		this.a = a;
		this.b = b;
		this.c = c;
	}

	static <A, B, C> UT<A, B, C> make(A a, B b, C c) {
		return new UT<A, B, C>(a, b, c);
	}

	public boolean equals(Object o) {
		if (o == this)
			return true;
		if (!(o instanceof UT))
			return false;
		UT<?, ?, ?> t = (UT<?, ?, ?>) o;
		boolean aok = a == null ? t.a == null : a.equals(t.a);
		boolean bok = b == null ? t.b == null : b.equals(t.b);
		boolean cok = c == null ? t.c == null : c.equals(t.c);
		return aok && bok && cok;
	}

	public String toString() {
		return "(" + a.toString() + ", " + b.toString() + ", " + c.toString() + ")";
	}
}

class T<A extends Comparable<? super A>, B extends Comparable<? super B>, C extends Comparable<? super C>>
		extends UT<A, B, C> implements Comparable<T<A, B, C>> { // Tuple
	T(A a, B b, C c) {
		super(a, b, c);
	}

	static <A extends Comparable<? super A>, B extends Comparable<? super B>, C extends Comparable<? super C>> T<A, B, C> make(
			A a, B b, C c) {
		return new T<A, B, C>(a, b, c);
	}

	public int compareTo(T<A, B, C> o) {
		int sa = a.compareTo(o.a);
		int sb = b.compareTo(o.b);
		int sc = c.compareTo(o.c);
		return sa != 0 ? sa : sb != 0 ? sb : sc;
	}
}

class TI implements Comparable<TI> { // Tuple int
	int a;
	int b;
	int c;

	TI(int a, int b, int c) {
		this.a = a;
		this.b = b;
		this.c = c;
	}

	static TI make(int a, int b, int c) {
		return new TI(a, b, c);
	}

	TL toLong() {
		return TL.make(a, b, c);
	}

	public boolean equals(Object o) {
		if (o == this)
			return true;
		if (!(o instanceof TI))
			return false;
		TI t = (TI) o;
		return a == t.a && b == t.b && c == t.c;
	}

	public int compareTo(TI o) {
		int sa = a - o.a;
		int sb = b - o.b;
		int sc = c - o.c;
		return sa > 0 ? 1 : sa < 0 ? -1 : sb > 0 ? 1 : sb < 0 ? -1 : sc > 0 ? 1 : sc < 0 ? -1 : 0;
	}

	public String toString() {
		return "(" + a + ", " + b + ", " + c + ")";
	}
}

class TL implements Comparable<TL> { // Tuple long
	long a;
	long b;
	long c;

	TL(long a, long b, long c) {
		this.a = a;
		this.b = b;
		this.c = c;
	}

	static TL make(long a, long b, long c) {
		return new TL(a, b, c);
	}

	TI toInt() {
		return TI.make((int) a, (int) b, (int) c);
	}

	public boolean equals(Object o) {
		if (o == this)
			return true;
		if (!(o instanceof TL))
			return false;
		TL t = (TL) o;
		return a == t.a && b == t.b && c == t.c;
	}

	public int compareTo(TL o) {
		long sa = a - o.a;
		long sb = b - o.b;
		long sc = c - o.c;
		return sa > 0 ? 1 : sa < 0 ? -1 : sb > 0 ? 1 : sb < 0 ? -1 : sc > 0 ? 1 : sc < 0 ? -1 : 0;
	}

	public String toString() {
		return "(" + a + ", " + b + ", " + c + ")";
	}
}

final class U { // Utilities
	private U() {
	}

	static <A> ArrayList<A> make(int n, F.IX<A> maker) {
		ArrayList<A> res = new ArrayList<A>();
		for (int i = 0; i < n; i++)
			res.add(maker.f(i));
		return res;
	}

	static boolean[] makeB(int n, F.IB maker) {
		boolean[] res = new boolean[n];
		for (int i = 0; i < n; i++)
			res[i] = maker.f(i);
		return res;
	}

	static int[] makeI(int n, F.II maker) {
		int[] res = new int[n];
		for (int i = 0; i < n; i++)
			res[i] = maker.f(i);
		return res;
	}

	static long[] makeL(int n, F.IL maker) {
		long[] res = new long[n];
		for (int i = 0; i < n; i++)
			res[i] = maker.f(i);
		return res;
	}

	static <A> A[] makeX(int n, F.IX<A> maker, A[] as) {
		A[] res = Arrays.copyOf(as, n);
		for (int i = 0; i < n; i++)
			res[i] = maker.f(i);
		return res;
	}

	static <A> ArrayList<A> filter(ArrayList<A> as, F.XB<A> pred) {
		ArrayList<A> res = new ArrayList<A>();
		for (A a : as)
			if (pred.f(a))
				res.add(a);
		return res;
	}

	static <A> int count(ArrayList<A> as, F.XB<A> pred) {
		int res = 0;
		for (A a : as)
			if (pred.f(a))
				res++;
		return res;
	}

	static <A> ArrayList<A> concat(ArrayList<A> as, ArrayList<A> bs) {
		ArrayList<A> res = new ArrayList<A>();
		res.addAll(as);
		res.addAll(bs);
		return res;
	}

	static <A> boolean any(ArrayList<A> as, F.XB<A> pred) {
		for (A a : as)
			if (pred.f(a))
				return true;
		return false;
	}

	static <A> boolean all(ArrayList<A> as, F.XB<A> pred) {
		for (A a : as)
			if (!pred.f(a))
				return false;
		return true;
	}

	static <A> ArrayList<A> flatten(ArrayList<ArrayList<A>> ass) {
		ArrayList<A> res = new ArrayList<A>();
		for (ArrayList<A> as : ass)
			res.addAll(as);
		return res;
	}

	static <A, B> B foldl(ArrayList<A> as, F.XXX<B, A, B> f, B e) {
		B res = e;
		for (A a : as)
			res = f.f(res, a);
		return res;
	}

	static <A, B> B foldr(ArrayList<A> as, F.XXX<A, B, B> f, B e) {
		B res = e;
		for (int i = as.size() - 1; i >= 0; i--)
			res = f.f(as.get(i), res);
		return res;
	}

	static <A> ArrayList<A> reverse(ArrayList<A> as) {
		int size = as.size();
		return make(size, i -> as.get(size - 1 - i));
	}

	static boolean[] reverse(boolean[] as) {
		int size = as.length;
		return makeB(size, i -> as[size - 1 - i]);
	}

	static int[] reverse(int[] as) {
		int size = as.length;
		return makeI(size, i -> as[size - 1 - i]);
	}

	static long[] reverse(long[] as) {
		int size = as.length;
		return makeL(size, i -> as[size - 1 - i]);
	}

	static <A> A[] reverse(A[] as) {
		int size = as.length;
		return makeX(size, i -> as[size - 1 - i], as);
	}

	static <A extends Comparable<? super A>> UP<TreeMap<A, Integer>, ArrayList<A>> compress(ArrayList<A> as) {
		TreeSet<A> set = new TreeSet<A>(as);
		TreeMap<A, Integer> map = new TreeMap<A, Integer>();
		ArrayList<A> imap = new ArrayList<A>();
		int i = 0;
		for (A a : set) {
			map.put(a, i++);
			imap.add(a);
		}
		return UP.make(map, imap);
	}

	static <A, B> ArrayList<B> map(ArrayList<A> as, F.XX<A, B> f) {
		return make(as.size(), (i) -> f.f(as.get(i)));
	}

	static <A, B> ArrayList<B> mapi(ArrayList<A> as, F.XIX<A, B> f) {
		return make(as.size(), (i) -> f.f(as.get(i), i));
	}

	static <A, B> ArrayList<UP<A, B>> zip(ArrayList<A> as, ArrayList<B> bs) {
		return make(min(as.size(), bs.size()), (i) -> UP.make(as.get(i), bs.get(i)));
	}

	static int min(int a, int b) {
		return a < b ? a : b;
	}

	static long min(long a, long b) {
		return a < b ? a : b;
	}

	static <A extends Comparable<? super A>> A min(A a, A b) {
		return a.compareTo(b) < 0 ? a : b;
	}

	static int max(int a, int b) {
		return a > b ? a : b;
	}

	static long max(long a, long b) {
		return a > b ? a : b;
	}

	static <A extends Comparable<? super A>> A max(A a, A b) {
		return a.compareTo(b) > 0 ? a : b;
	}

	static int clamp(int a, int min, int max) {
		return a < min ? min : a > max ? max : a;
	}

	static long clamp(long a, long min, long max) {
		return a < min ? min : a > max ? max : a;
	}

	static <A extends Comparable<? super A>> A clamp(A a, A min, A max) {
		return a.compareTo(min) < 0 ? min : a.compareTo(max) > 0 ? max : a;
	}

	static int abs(int a) {
		return a < 0 ? -a : a;
	}

	static long abs(long a) {
		return a < 0 ? -a : a;
	}

	static void forEachBitPerm(int n, int k, F.IV f) {
		for (int i = (1 << k) - 1; i < 1 << n;) {
			f.f(i);
			int t = (i | i - 1) + 1;
			i = t | ((t & -t) / (i & -i) >> 1) - 1;
		}
	}

	static int nextBitPerm(int a) {
		int t = (a | a - 1) + 1;
		return t | ((t & -t) / (a & -a) >> 1) - 1;
	}

	static void mebius(int n, F.IIV f) { // s, i
		int bit = 1;
		int exp = 0;
		for (int i = 1; i < 1 << n; i++) {
			f.f(i ^ bit, exp);
			if ((i & i + 1) == 0) {
				bit <<= 1;
				exp++;
			}
		}
	}

	static void zeta(int n, F.IIV f) { // s, i
		int m = (1 << n) - 1;
		int bit = 1;
		int exp = 0;
		for (int i = (1 << n) - 2; i >= 0; i--) {
			f.f(i ^ bit, exp);
			if ((~i & ~i + 1 & m) == 0) {
				bit <<= 1;
				exp++;
			}
		}
	}

	static <A> ArrayList<A> toAL(A[] as) {
		return make(as.length, i -> as[i]);
	}

	static <A> A[] doubleSize(A[] as) {
		return Arrays.copyOf(as, as.length << 1);
	}

	static long searchL(long ng, long ok, F.LB isOk) {
		while (ng - ok > 1 || ok - ng > 1) {
			long mid = ng + ok >> 1;
			if (isOk.f(mid))
				ok = mid;
			else
				ng = mid;
		}
		return ok;
	}

	static int searchI(int ng, int ok, F.IB isOk) {
		return (int) searchL((long) ng, (long) ok, (mid) -> isOk.f((int) mid));
	}
}

final class F { // Functions
	private F() {
	}

	interface VV {
		void f();
	}

	interface BV {
		void f(boolean a);
	}

	interface BXV<A> {
		void f(boolean a, A b);
	}

	interface BXXV<A, B> {
		void f(boolean a, A b, B c);
	}

	interface BXXXV<A, B, C> {
		void f(boolean a, A b, B c, C d);
	}

	interface XBV<A> {
		void f(A a, boolean b);
	}

	interface XXBV<A, B> {
		void f(A a, B b, boolean c);
	}

	interface XXXBV<A, B, C> {
		void f(A a, B b, C c, boolean d);
	}

	interface IV {
		void f(int a);
	}

	interface IXV<A> {
		void f(int a, A b);
	}

	interface IXXV<A, B> {
		void f(int a, A b, B c);
	}

	interface IXXXV<A, B, C> {
		void f(int a, A b, B c, C d);
	}

	interface XIV<A> {
		void f(A a, int b);
	}

	interface XXIV<A, B> {
		void f(A a, B b, int c);
	}

	interface XXXIV<A, B, C> {
		void f(A a, B b, C c, int d);
	}

	interface LV {
		void f(long a);
	}

	interface LXV<A> {
		void f(long a, A b);
	}

	interface LXXV<A, B> {
		void f(long a, A b, B c);
	}

	interface LXXXV<A, B, C> {
		void f(long a, A b, B c, C d);
	}

	interface XLV<A> {
		void f(A a, long b);
	}

	interface XXLV<A, B> {
		void f(A a, B b, long c);
	}

	interface XXXLV<A, B, C> {
		void f(A a, B b, C c, long d);
	}

	interface DV {
		void f(double a);
	}

	interface DXV<A> {
		void f(double a, A b);
	}

	interface DXXV<A, B> {
		void f(double a, A b, B c);
	}

	interface DXXXV<A, B, C> {
		void f(double a, A b, B c, C d);
	}

	interface XDV<A> {
		void f(A a, double b);
	}

	interface XXDV<A, B> {
		void f(A a, B b, double c);
	}

	interface XXXDV<A, B, C> {
		void f(A a, B b, C c, double d);
	}

	interface XV<A> {
		void f(A a);
	}

	interface XXV<A, B> {
		void f(A a, B b);
	}

	interface XXXV<A, B, C> {
		void f(A a, B b, C c);
	}

	interface XXXXV<A, B, C, D> {
		void f(A a, B b, C c, D d);
	}

	interface BBV {
		void f(boolean a, boolean b);
	}

	interface BIV {
		void f(boolean a, int b);
	}

	interface BLV {
		void f(boolean a, long b);
	}

	interface BDV {
		void f(boolean a, double b);
	}

	interface IBV {
		void f(int a, boolean b);
	}

	interface IIV {
		void f(int a, int b);
	}

	interface ILV {
		void f(int a, long b);
	}

	interface IDV {
		void f(int a, double b);
	}

	interface LBV {
		void f(long a, boolean b);
	}

	interface LIV {
		void f(long a, int b);
	}

	interface LLV {
		void f(long a, long b);
	}

	interface LDV {
		void f(long a, double b);
	}

	interface DBV {
		void f(double a, boolean b);
	}

	interface DIV {
		void f(double a, int b);
	}

	interface DLV {
		void f(double a, long b);
	}

	interface DDV {
		void f(double a, double b);
	}

	interface VB {
		boolean f();
	}

	interface BB {
		boolean f(boolean a);
	}

	interface BXB<A> {
		boolean f(boolean a, A b);
	}

	interface BXXB<A, B> {
		boolean f(boolean a, A b, B c);
	}

	interface BXXXB<A, B, C> {
		boolean f(boolean a, A b, B c, C d);
	}

	interface XBB<A> {
		boolean f(A a, boolean b);
	}

	interface XXBB<A, B> {
		boolean f(A a, B b, boolean c);
	}

	interface XXXBB<A, B, C> {
		boolean f(A a, B b, C c, boolean d);
	}

	interface IB {
		boolean f(int a);
	}

	interface IXB<A> {
		boolean f(int a, A b);
	}

	interface IXXB<A, B> {
		boolean f(int a, A b, B c);
	}

	interface IXXXB<A, B, C> {
		boolean f(int a, A b, B c, C d);
	}

	interface XIB<A> {
		boolean f(A a, int b);
	}

	interface XXIB<A, B> {
		boolean f(A a, B b, int c);
	}

	interface XXXIB<A, B, C> {
		boolean f(A a, B b, C c, int d);
	}

	interface LB {
		boolean f(long a);
	}

	interface LXB<A> {
		boolean f(long a, A b);
	}

	interface LXXB<A, B> {
		boolean f(long a, A b, B c);
	}

	interface LXXXB<A, B, C> {
		boolean f(long a, A b, B c, C d);
	}

	interface XLB<A> {
		boolean f(A a, long b);
	}

	interface XXLB<A, B> {
		boolean f(A a, B b, long c);
	}

	interface XXXLB<A, B, C> {
		boolean f(A a, B b, C c, long d);
	}

	interface DB {
		boolean f(double a);
	}

	interface DXB<A> {
		boolean f(double a, A b);
	}

	interface DXXB<A, B> {
		boolean f(double a, A b, B c);
	}

	interface DXXXB<A, B, C> {
		boolean f(double a, A b, B c, C d);
	}

	interface XDB<A> {
		boolean f(A a, double b);
	}

	interface XXDB<A, B> {
		boolean f(A a, B b, double c);
	}

	interface XXXDB<A, B, C> {
		boolean f(A a, B b, C c, double d);
	}

	interface XB<A> {
		boolean f(A a);
	}

	interface XXB<A, B> {
		boolean f(A a, B b);
	}

	interface XXXB<A, B, C> {
		boolean f(A a, B b, C c);
	}

	interface XXXXB<A, B, C, D> {
		boolean f(A a, B b, C c, D d);
	}

	interface BBB {
		boolean f(boolean a, boolean b);
	}

	interface BIB {
		boolean f(boolean a, int b);
	}

	interface BLB {
		boolean f(boolean a, long b);
	}

	interface BDB {
		boolean f(boolean a, double b);
	}

	interface IBB {
		boolean f(int a, boolean b);
	}

	interface IIB {
		boolean f(int a, int b);
	}

	interface ILB {
		boolean f(int a, long b);
	}

	interface IDB {
		boolean f(int a, double b);
	}

	interface LBB {
		boolean f(long a, boolean b);
	}

	interface LIB {
		boolean f(long a, int b);
	}

	interface LLB {
		boolean f(long a, long b);
	}

	interface LDB {
		boolean f(long a, double b);
	}

	interface DBB {
		boolean f(double a, boolean b);
	}

	interface DIB {
		boolean f(double a, int b);
	}

	interface DLB {
		boolean f(double a, long b);
	}

	interface DDB {
		boolean f(double a, double b);
	}

	interface VI {
		int f();
	}

	interface BI {
		int f(boolean a);
	}

	interface BXI<A> {
		int f(boolean a, A b);
	}

	interface BXXI<A, B> {
		int f(boolean a, A b, B c);
	}

	interface BXXXI<A, B, C> {
		int f(boolean a, A b, B c, C d);
	}

	interface XBI<A> {
		int f(A a, boolean b);
	}

	interface XXBI<A, B> {
		int f(A a, B b, boolean c);
	}

	interface XXXBI<A, B, C> {
		int f(A a, B b, C c, boolean d);
	}

	interface II {
		int f(int a);
	}

	interface IXI<A> {
		int f(int a, A b);
	}

	interface IXXI<A, B> {
		int f(int a, A b, B c);
	}

	interface IXXXI<A, B, C> {
		int f(int a, A b, B c, C d);
	}

	interface XII<A> {
		int f(A a, int b);
	}

	interface XXII<A, B> {
		int f(A a, B b, int c);
	}

	interface XXXII<A, B, C> {
		int f(A a, B b, C c, int d);
	}

	interface LI {
		int f(long a);
	}

	interface LXI<A> {
		int f(long a, A b);
	}

	interface LXXI<A, B> {
		int f(long a, A b, B c);
	}

	interface LXXXI<A, B, C> {
		int f(long a, A b, B c, C d);
	}

	interface XLI<A> {
		int f(A a, long b);
	}

	interface XXLI<A, B> {
		int f(A a, B b, long c);
	}

	interface XXXLI<A, B, C> {
		int f(A a, B b, C c, long d);
	}

	interface DI {
		int f(double a);
	}

	interface DXI<A> {
		int f(double a, A b);
	}

	interface DXXI<A, B> {
		int f(double a, A b, B c);
	}

	interface DXXXI<A, B, C> {
		int f(double a, A b, B c, C d);
	}

	interface XDI<A> {
		int f(A a, double b);
	}

	interface XXDI<A, B> {
		int f(A a, B b, double c);
	}

	interface XXXDI<A, B, C> {
		int f(A a, B b, C c, double d);
	}

	interface XI<A> {
		int f(A a);
	}

	interface XXI<A, B> {
		int f(A a, B b);
	}

	interface XXXI<A, B, C> {
		int f(A a, B b, C c);
	}

	interface XXXXI<A, B, C, D> {
		int f(A a, B b, C c, D d);
	}

	interface BBI {
		int f(boolean a, boolean b);
	}

	interface BII {
		int f(boolean a, int b);
	}

	interface BLI {
		int f(boolean a, long b);
	}

	interface BDI {
		int f(boolean a, double b);
	}

	interface IBI {
		int f(int a, boolean b);
	}

	interface III {
		int f(int a, int b);
	}

	interface ILI {
		int f(int a, long b);
	}

	interface IDI {
		int f(int a, double b);
	}

	interface LBI {
		int f(long a, boolean b);
	}

	interface LII {
		int f(long a, int b);
	}

	interface LLI {
		int f(long a, long b);
	}

	interface LDI {
		int f(long a, double b);
	}

	interface DBI {
		int f(double a, boolean b);
	}

	interface DII {
		int f(double a, int b);
	}

	interface DLI {
		int f(double a, long b);
	}

	interface DDI {
		int f(double a, double b);
	}

	interface VL {
		long f();
	}

	interface BL {
		long f(boolean a);
	}

	interface BXL<A> {
		long f(boolean a, A b);
	}

	interface BXXL<A, B> {
		long f(boolean a, A b, B c);
	}

	interface BXXXL<A, B, C> {
		long f(boolean a, A b, B c, C d);
	}

	interface XBL<A> {
		long f(A a, boolean b);
	}

	interface XXBL<A, B> {
		long f(A a, B b, boolean c);
	}

	interface XXXBL<A, B, C> {
		long f(A a, B b, C c, boolean d);
	}

	interface IL {
		long f(int a);
	}

	interface IXL<A> {
		long f(int a, A b);
	}

	interface IXXL<A, B> {
		long f(int a, A b, B c);
	}

	interface IXXXL<A, B, C> {
		long f(int a, A b, B c, C d);
	}

	interface XIL<A> {
		long f(A a, int b);
	}

	interface XXIL<A, B> {
		long f(A a, B b, int c);
	}

	interface XXXIL<A, B, C> {
		long f(A a, B b, C c, int d);
	}

	interface LL {
		long f(long a);
	}

	interface LXL<A> {
		long f(long a, A b);
	}

	interface LXXL<A, B> {
		long f(long a, A b, B c);
	}

	interface LXXXL<A, B, C> {
		long f(long a, A b, B c, C d);
	}

	interface XLL<A> {
		long f(A a, long b);
	}

	interface XXLL<A, B> {
		long f(A a, B b, long c);
	}

	interface XXXLL<A, B, C> {
		long f(A a, B b, C c, long d);
	}

	interface DL {
		long f(double a);
	}

	interface DXL<A> {
		long f(double a, A b);
	}

	interface DXXL<A, B> {
		long f(double a, A b, B c);
	}

	interface DXXXL<A, B, C> {
		long f(double a, A b, B c, C d);
	}

	interface XDL<A> {
		long f(A a, double b);
	}

	interface XXDL<A, B> {
		long f(A a, B b, double c);
	}

	interface XXXDL<A, B, C> {
		long f(A a, B b, C c, double d);
	}

	interface XL<A> {
		long f(A a);
	}

	interface XXL<A, B> {
		long f(A a, B b);
	}

	interface XXXL<A, B, C> {
		long f(A a, B b, C c);
	}

	interface XXXXL<A, B, C, D> {
		long f(A a, B b, C c, D d);
	}

	interface BBL {
		long f(boolean a, boolean b);
	}

	interface BIL {
		long f(boolean a, int b);
	}

	interface BLL {
		long f(boolean a, long b);
	}

	interface BDL {
		long f(boolean a, double b);
	}

	interface IBL {
		long f(int a, boolean b);
	}

	interface IIL {
		long f(int a, int b);
	}

	interface ILL {
		long f(int a, long b);
	}

	interface IDL {
		long f(int a, double b);
	}

	interface LBL {
		long f(long a, boolean b);
	}

	interface LIL {
		long f(long a, int b);
	}

	interface LLL {
		long f(long a, long b);
	}

	interface LDL {
		long f(long a, double b);
	}

	interface DBL {
		long f(double a, boolean b);
	}

	interface DIL {
		long f(double a, int b);
	}

	interface DLL {
		long f(double a, long b);
	}

	interface DDL {
		long f(double a, double b);
	}

	interface VD {
		double f();
	}

	interface BD {
		double f(boolean a);
	}

	interface BXD<A> {
		double f(boolean a, A b);
	}

	interface BXXD<A, B> {
		double f(boolean a, A b, B c);
	}

	interface BXXXD<A, B, C> {
		double f(boolean a, A b, B c, C d);
	}

	interface XBD<A> {
		double f(A a, boolean b);
	}

	interface XXBD<A, B> {
		double f(A a, B b, boolean c);
	}

	interface XXXBD<A, B, C> {
		double f(A a, B b, C c, boolean d);
	}

	interface ID {
		double f(int a);
	}

	interface IXD<A> {
		double f(int a, A b);
	}

	interface IXXD<A, B> {
		double f(int a, A b, B c);
	}

	interface IXXXD<A, B, C> {
		double f(int a, A b, B c, C d);
	}

	interface XID<A> {
		double f(A a, int b);
	}

	interface XXID<A, B> {
		double f(A a, B b, int c);
	}

	interface XXXID<A, B, C> {
		double f(A a, B b, C c, int d);
	}

	interface LD {
		double f(long a);
	}

	interface LXD<A> {
		double f(long a, A b);
	}

	interface LXXD<A, B> {
		double f(long a, A b, B c);
	}

	interface LXXXD<A, B, C> {
		double f(long a, A b, B c, C d);
	}

	interface XLD<A> {
		double f(A a, long b);
	}

	interface XXLD<A, B> {
		double f(A a, B b, long c);
	}

	interface XXXLD<A, B, C> {
		double f(A a, B b, C c, long d);
	}

	interface DD {
		double f(double a);
	}

	interface DXD<A> {
		double f(double a, A b);
	}

	interface DXXD<A, B> {
		double f(double a, A b, B c);
	}

	interface DXXXD<A, B, C> {
		double f(double a, A b, B c, C d);
	}

	interface XDD<A> {
		double f(A a, double b);
	}

	interface XXDD<A, B> {
		double f(A a, B b, double c);
	}

	interface XXXDD<A, B, C> {
		double f(A a, B b, C c, double d);
	}

	interface XD<A> {
		double f(A a);
	}

	interface XXD<A, B> {
		double f(A a, B b);
	}

	interface XXXD<A, B, C> {
		double f(A a, B b, C c);
	}

	interface XXXXD<A, B, C, D> {
		double f(A a, B b, C c, D d);
	}

	interface BBD {
		double f(boolean a, boolean b);
	}

	interface BID {
		double f(boolean a, int b);
	}

	interface BLD {
		double f(boolean a, long b);
	}

	interface BDD {
		double f(boolean a, double b);
	}

	interface IBD {
		double f(int a, boolean b);
	}

	interface IID {
		double f(int a, int b);
	}

	interface ILD {
		double f(int a, long b);
	}

	interface IDD {
		double f(int a, double b);
	}

	interface LBD {
		double f(long a, boolean b);
	}

	interface LID {
		double f(long a, int b);
	}

	interface LLD {
		double f(long a, long b);
	}

	interface LDD {
		double f(long a, double b);
	}

	interface DBD {
		double f(double a, boolean b);
	}

	interface DID {
		double f(double a, int b);
	}

	interface DLD {
		double f(double a, long b);
	}

	interface DDD {
		double f(double a, double b);
	}

	interface VX<A> {
		A f();
	}

	interface BX<A> {
		A f(boolean a);
	}

	interface BXX<A, B> {
		B f(boolean a, A b);
	}

	interface BXXX<A, B, C> {
		C f(boolean a, A b, B c);
	}

	interface BXXXX<A, B, C, D> {
		D f(boolean a, A b, B c, C d);
	}

	interface XBX<A, B> {
		B f(A a, boolean b);
	}

	interface XXBX<A, B, C> {
		C f(A a, B b, boolean c);
	}

	interface XXXBX<A, B, C, D> {
		D f(A a, B b, C c, boolean d);
	}

	interface IX<A> {
		A f(int a);
	}

	interface IXX<A, B> {
		B f(int a, A b);
	}

	interface IXXX<A, B, C> {
		C f(int a, A b, B c);
	}

	interface IXXXX<A, B, C, D> {
		D f(int a, A b, B c, C d);
	}

	interface XIX<A, B> {
		B f(A a, int b);
	}

	interface XXIX<A, B, C> {
		C f(A a, B b, int c);
	}

	interface XXXIX<A, B, C, D> {
		D f(A a, B b, C c, int d);
	}

	interface LX<A> {
		A f(long a);
	}

	interface LXX<A, B> {
		B f(long a, A b);
	}

	interface LXXX<A, B, C> {
		C f(long a, A b, B c);
	}

	interface LXXXX<A, B, C, D> {
		D f(long a, A b, B c, C d);
	}

	interface XLX<A, B> {
		B f(A a, long b);
	}

	interface XXLX<A, B, C> {
		C f(A a, B b, long c);
	}

	interface XXXLX<A, B, C, D> {
		D f(A a, B b, C c, long d);
	}

	interface DX<A> {
		A f(double a);
	}

	interface DXX<A, B> {
		B f(double a, A b);
	}

	interface DXXX<A, B, C> {
		C f(double a, A b, B c);
	}

	interface DXXXX<A, B, C, D> {
		D f(double a, A b, B c, C d);
	}

	interface XDX<A, B> {
		B f(A a, double b);
	}

	interface XXDX<A, B, C> {
		C f(A a, B b, double c);
	}

	interface XXXDX<A, B, C, D> {
		D f(A a, B b, C c, double d);
	}

	interface XX<A, B> {
		B f(A a);
	}

	interface XXX<A, B, C> {
		C f(A a, B b);
	}

	interface XXXX<A, B, C, D> {
		D f(A a, B b, C c);
	}

	interface XXXXX<A, B, C, D, E> {
		E f(A a, B b, C c, D d);
	}

	interface BBX<A> {
		A f(boolean a, boolean b);
	}

	interface BIX<A> {
		A f(boolean a, int b);
	}

	interface BLX<A> {
		A f(boolean a, long b);
	}

	interface BDX<A> {
		A f(boolean a, double b);
	}

	interface IBX<A> {
		A f(int a, boolean b);
	}

	interface IIX<A> {
		A f(int a, int b);
	}

	interface ILX<A> {
		A f(int a, long b);
	}

	interface IDX<A> {
		A f(int a, double b);
	}

	interface LBX<A> {
		A f(long a, boolean b);
	}

	interface LIX<A> {
		A f(long a, int b);
	}

	interface LLX<A> {
		A f(long a, long b);
	}

	interface LDX<A> {
		A f(long a, double b);
	}

	interface DBX<A> {
		A f(double a, boolean b);
	}

	interface DIX<A> {
		A f(double a, int b);
	}

	interface DLX<A> {
		A f(double a, long b);
	}

	interface DDX<A> {
		A f(double a, double b);
	}
}

class SA { // suffix array
	static int[] makeSA(String s) {
		int n = s.length() + 1;
		int[] cs = new int[n];
		cs[n - 1] = 0;
		for (int i = 0; i < n - 1; i++)
			cs[i] = s.charAt(i) + 1;
		ArrayList<Integer> acs = U.make(n, i -> cs[i]);
		TreeMap<Integer, Integer> tm = U.compress(acs).a;
		int k = tm.size();
		for (int i = 0; i < n; i++)
			cs[i] = tm.get(cs[i]);
		return makeSA(cs, n, k);
	}

	static int[] makeLCP(String s, int[] sa) { // lcp(i, i+1)
		int n = sa.length;
		int[] r = new int[n];
		for (int i = 0; i < n; i++)
			r[sa[i]] = i;
		int[] lcp = new int[n];
		int l = 0;
		for (int i = 0; i < n; i++) {
			int idx = r[i];
			if (idx == n - 1) {
				lcp[idx] = -1;
				l = 0;
				continue;
			}
			int p = sa[idx];
			int q = sa[idx + 1];
			if (l > 0)
				l--;
			while (p + l < n - 1 && q + l < n - 1 && s.charAt(p + l) == s.charAt(q + l))
				l++;
			lcp[idx] = l;
		}
		return lcp;
	}

	static F.III lcpQuery(String s) {
		int n = s.length() + 1;
		int[] sa = makeSA(s);
		int[] lcp = makeLCP(s, sa);
		int[] inv = new int[n];
		for (int i = 0; i < n; i++) {
			inv[sa[i]] = i;
		}
		ST<Integer> st = new ST<Integer>(n, (a, b) -> a < b ? a : b, Integer.MAX_VALUE);
		st.init(i -> i < n ? lcp[i] : Integer.MAX_VALUE);
		return (i, j) -> {
			if (i == j)
				return n - 1 - i;
			i = inv[i];
			j = inv[j];
			if (i > j) {
				i ^= j;
				j ^= i;
				i ^= j;
			}
			return st.query(i, j);
		};
	}

	private static int[] makeSA(int[] cs, int n, int k) {
		boolean[] isS = new boolean[n];
		boolean[] isLms = new boolean[n];
		int[] lmss = new int[n];
		int numLmss = 0;
		isS[n - 1] = true;
		for (int i = n - 2; i >= 0; i--)
			isS[i] = cs[i] < cs[i + 1] || cs[i] == cs[i + 1] && isS[i + 1];
		for (int i = 1; i < n; i++)
			if (!isS[i - 1] && isS[i]) {
				lmss[numLmss++] = i;
				isLms[i] = true;
			}
		int[] sa = inducedSort(cs, n, numLmss, k, lmss, isS);
		int[] lmss2 = new int[numLmss];
		numLmss = 0;
		for (int i = 0; i < n; i++)
			if (isLms[sa[i]])
				lmss2[numLmss++] = sa[i];
		int num = 0;
		sa[lmss2[0]] = 0;
		for (int i = 0; i < numLmss - 1; i++) {
			int p = lmss2[i];
			int q = lmss2[i + 1];
			for (int j = 0; j < n; j++) {
				if (cs[p] != cs[q] || isLms[p] != isLms[q]) {
					sa[lmss2[i + 1]] = ++num;
					break;
				} else if (j > 0 && (isLms[p] || isLms[q])) {
					sa[lmss2[i + 1]] = num;
					break;
				}
				p++;
				q++;
			}
		}
		if (num + 1 < numLmss) {
			numLmss = 0;
			for (int i = 0; i < n; i++)
				if (isLms[i])
					lmss2[numLmss++] = sa[i];
			lmss2 = makeSA(lmss2, numLmss, num + 1);
			for (int i = 0; i < numLmss; i++)
				lmss2[i] = lmss[lmss2[i]];
		}
		return inducedSort(cs, n, numLmss, k, lmss2, isS);
	}

	private static int[] inducedSort(int[] cs, int n, int numLmss, int k, int[] lmss, boolean[] isS) {
		int[] sa = new int[n];
		int[] bin = new int[k + 1];
		for (int i = 0; i < n; i++)
			bin[cs[i] + 1]++;
		for (int i = 0; i < k; i++)
			bin[i + 1] += bin[i];
		int[] counts = new int[k];
		for (int i = numLmss - 1; i >= 0; i--) { // put LMS backward
			int c = cs[lmss[i]];
			sa[bin[c + 1] - 1 - counts[c]++] = lmss[i];
		}
		for (int i = 0; i < k; i++)
			counts[i] = 0;
		for (int i = 0; i < n; i++) { // put L forward
			int s = sa[i] - 1;
			if (s < 0 || isS[s])
				continue;
			int c = cs[s];
			sa[bin[c] + counts[c]++] = s;
		}
		for (int i = 0; i < k; i++)
			counts[i] = 0;
		for (int i = n - 1; i >= 0; i--) { // put S backward
			int s = sa[i] - 1;
			if (s < 0 || !isS[s])
				continue;
			int c = cs[s];
			sa[bin[c + 1] - 1 - counts[c]++] = s;
		}
		return sa;
	}
}

class ST<A> { // Segment Tree
	private ArrayList<A> as;
	private int h;
	private int n;
	private F.XXX<A, A, A> merger;
	private A e;

	ST(int num, F.XXX<A, A, A> merger, A e) {
		this.merger = merger;
		this.e = e;
		h = 0;
		while ((1 << h) < num)
			h++;
		n = 1 << h;
		as = U.make(2 * n, i -> e);
	}

	void init(A a) {
		init(i -> a);
	}

	void init(A[] as) {
		init(i -> i < as.length ? as[i] : e);
	}

	void init(F.IX<A> maker) {
		for (int i = 0; i < n; i++)
			as.set(n + i, maker.f(i));
		for (int i = n - 1; i > 0; i--)
			as.set(i, merge(as.get(i << 1), as.get(i << 1 | 1)));
	}

	A get(int i) {
		return query(i, i + 1);
	}

	void set(int i, A a) {
		as.set(i += n, a);
		while ((i >>= 1) > 0)
			as.set(i, merge(as.get(i << 1), as.get(i << 1 | 1)));
	}

	A query(int l, int r) {
		l += n;
		r += n;
		A al = e;
		A ar = e;
		while (l < r) {
			if ((l & 1) != 0)
				al = merge(al, as.get(l++));
			if ((r & 1) != 0)
				ar = merge(as.get(--r), ar);
			l >>= 1;
			r >>= 1;
		}
		return merge(al, ar);
	}

	private A merge(A a, A b) {
		return a == e ? b : b == e ? a : merger.f(a, b);
	}
}

class LST<A, Op> { // Lazy Segment Tree
	private ArrayList<A> as;
	private ArrayList<Op> lazy;
	private F.XXX<A, A, A> merger;
	private F.XXIX<A, Op, A> applier;
	private F.XXX<Op, Op, Op> opMerger;
	private A e;
	private Op id;
	private int h;
	private int n;

	LST(int num, F.XXX<A, A, A> merger, F.XXIX<A, Op, A> applier, F.XXX<Op, Op, Op> opMerger, A e, Op id) {
		this.merger = merger;
		this.applier = applier;
		this.opMerger = opMerger;
		this.e = e;
		this.id = id;
		h = 0;
		while ((1 << h) < num)
			h++;
		n = 1 << h;
		int size = n << 1;
		as = U.make(size, i -> e);
		lazy = U.make(size, i -> id);
	}

	void init(A a) {
		init(i -> a);
	}

	void init(A[] as) {
		init(i -> i < as.length ? as[i] : e);
	}

	void init(F.IX<A> maker) {
		for (int i = 0; i < n << 1; i++)
			lazy.set(i, id);
		for (int i = 0; i < n; i++)
			as.set(n + i, maker.f(i));
		for (int i = n - 1; i > 0; i--)
			as.set(i, merge(as.get(i << 1), as.get(i << 1 | 1)));
	}

	A get(int i) {
		return query(i, i + 1);
	}

	void set(int i, A a) {
		prop(i += n);
		as.set(i, a);
		lazy.set(i, id);
		backprop(i);
	}

	A query(int l, int r) {
		prop(l += n);
		prop(r += n - 1);
		r++;
		A al = e;
		A ar = e;
		int len = 1;
		while (l < r) {
			if ((l & 1) != 0)
				al = merge(al, eval(l++, len));
			if ((r & 1) != 0)
				ar = merge(eval(--r, len), ar);
			l >>= 1;
			r >>= 1;
			len <<= 1;
		}
		return merge(al, ar);
	}

	void apply(int l, int r, Op o) {
		prop(l += n);
		prop(r += n - 1);
		int a = l;
		int b = r;
		r++;
		while (l < r) {
			if ((l & 1) != 0)
				lazy.set(l, mergeOp(lazy.get(l++), o));
			if ((r & 1) != 0)
				lazy.set(--r, mergeOp(lazy.get(r), o));
			l >>= 1;
			r >>= 1;
		}
		backprop(a);
		backprop(b);
	}

	int size() {
		return n;
	}

	private A merge(A a, A b) {
		return a == e ? b : b == e ? a : merger.f(a, b);
	}

	private Op mergeOp(Op o, Op p) {
		return o == id ? p : p == id ? o : opMerger.f(o, p);
	}

	private A apply(A a, Op o, int len) {
		return o == id ? a : applier.f(a, o, len);
	}

	private A eval(int k, int len) {
		return apply(as.get(k), lazy.get(k), len);
	}

	private void flush(int k, int len) {
		Op o = lazy.get(k);
		if (o == id)
			return;
		lazy.set(k << 1, mergeOp(lazy.get(k << 1), o));
		lazy.set(k << 1 | 1, mergeOp(lazy.get(k << 1 | 1), o));
		as.set(k, apply(as.get(k), o, len));
		lazy.set(k, id);
	}

	private void prop(int k) {
		for (int i = h; i > 0; i--)
			flush(k >> i, 1 << i);
	}

	private void backprop(int k) {
		int len = 1;
		while ((k >>= 1) > 0) {
			as.set(k, merge(eval(k << 1, len), eval(k << 1 | 1, len)));
			len <<= 1;
		}
	}
}

interface Magma<A> {
	A g(A a, A b);
}

interface Associative {
}

interface Unital<A> {
	A e();
}

interface Invertible<A> {
	A inv(A a);
}

interface Commutative {
}

interface SemiGroup<A> extends Magma<A>, Associative {
}

interface Monoid<A> extends SemiGroup<A>, Unital<A> {
	static <A> Monoid<A> make(F.XXX<A, A, A> g, A e) {
		return new Monoid<A>() {
			public A g(A a, A b) {
				return a.equals(e) ? b : b.equals(e) ? a : g.f(a, b);
			}

			public A e() {
				return e;
			}
		};
	}
}

interface CommutativeMonoid<A> extends Monoid<A>, Commutative {
	static <A> CommutativeMonoid<A> make(F.XXX<A, A, A> g, A e) {
		return new CommutativeMonoid<A>() {
			public A g(A a, A b) {
				return a.equals(e) ? b : b.equals(e) ? a : g.f(a, b);
			}

			public A e() {
				return e;
			}
		};
	}
}

interface Group<A> extends Monoid<A>, Invertible<A> {
	static <A> Group<A> make(F.XXX<A, A, A> g, F.XX<A, A> inv, A e) {
		return new Group<A>() {
			public A g(A a, A b) {
				return a.equals(e) ? b : b.equals(e) ? a : g.f(a, b);
			}

			public A e() {
				return e;
			}

			public A inv(A a) {
				return a.equals(e) ? e : inv.f(a);
			}
		};
	}
}

interface AbelianGroup<A> extends Group<A>, CommutativeMonoid<A> {
	static <A> AbelianGroup<A> make(F.XXX<A, A, A> g, F.XX<A, A> inv, A e) {
		return new AbelianGroup<A>() {
			public A g(A a, A b) {
				return a.equals(e) ? b : b.equals(e) ? a : g.f(a, b);
			}

			public A e() {
				return e;
			}

			public A inv(A a) {
				return a.equals(e) ? e : inv.f(a);
			}
		};
	}
}

interface Ring<A> {
	AbelianGroup<A> add();

	Monoid<A> mul();

	default A zero() {
		return add().e();
	}

	default A one() {
		return mul().e();
	}

	static <A> Ring<A> make(F.XXX<A, A, A> add, F.XX<A, A> neg, F.XXX<A, A, A> mul, A zero, A one) {
		return make(AbelianGroup.make(add, neg, zero), Monoid.make(mul, one));
	}

	static <A> Ring<A> make(AbelianGroup<A> add, Monoid<A> mul) {
		return new Ring<A>() {
			public AbelianGroup<A> add() {
				return add;
			}

			public Monoid<A> mul() {
				return mul;
			}
		};
	}
}

interface CommutativeRing<A> extends Ring<A> {
	CommutativeMonoid<A> mul();

	static <A> CommutativeRing<A> make(F.XXX<A, A, A> add, F.XX<A, A> neg, F.XXX<A, A, A> mul, A zero, A one) {
		return make(AbelianGroup.make(add, neg, zero), CommutativeMonoid.make(mul, one));
	}

	static <A> CommutativeRing<A> make(AbelianGroup<A> add, CommutativeMonoid<A> mul) {
		return new CommutativeRing<A>() {
			public AbelianGroup<A> add() {
				return add;
			}

			public CommutativeMonoid<A> mul() {
				return mul;
			}
		};
	}
}

interface EuclideanRing<A> extends CommutativeRing<A> {
	A div(A a, A b);

	A mod(A a, A b);

	static <A> EuclideanRing<A> make(F.XXX<A, A, A> add, F.XX<A, A> neg, F.XXX<A, A, A> mul, F.XXX<A, A, A> div,
			F.XXX<A, A, A> mod, A zero, A one) {
		return make(AbelianGroup.make(add, neg, zero), CommutativeMonoid.make(mul, one), div, mod);
	}

	static <A> EuclideanRing<A> make(AbelianGroup<A> add, CommutativeMonoid<A> mul, F.XXX<A, A, A> div,
			F.XXX<A, A, A> mod) {
		final A zero = add.e();
		final A one = mul.e();
		return new EuclideanRing<A>() {
			public AbelianGroup<A> add() {
				return add;
			}

			public CommutativeMonoid<A> mul() {
				return mul;
			}

			public A div(A a, A b) {
				if (b.equals(zero))
					throw new ArithmeticException("division by zero");
				return b.equals(one) ? a : div.f(a, b);
			}

			public A mod(A a, A b) {
				if (b.equals(zero))
					throw new ArithmeticException("division by zero");
				return b.equals(one) ? zero : mod.f(a, b);
			}
		};
	}
}

interface Field<A> extends EuclideanRing<A> {
	AbelianGroup<A> mul();

	static <A> Field<A> make(F.XXX<A, A, A> add, F.XX<A, A> neg, F.XXX<A, A, A> mul, F.XX<A, A> inv, A zero, A one) {
		return make(AbelianGroup.make(add, neg, zero), AbelianGroup.make(mul, inv, one));
	}

	static <A> Field<A> make(AbelianGroup<A> add, AbelianGroup<A> mul) {
		final A zero = add.e();
		final A one = mul.e();
		return new Field<A>() {
			public AbelianGroup<A> add() {
				return add;
			}

			public AbelianGroup<A> mul() {
				return mul;
			}

			public A div(A a, A b) {
				if (b.equals(zero))
					throw new ArithmeticException("division by zero");
				return b.equals(one) ? a : mul.g(a, mul.inv(b));
			}

			public A mod(A a, A b) {
				if (b.equals(zero))
					throw new ArithmeticException("division by zero");
				return zero;
			}
		};
	}
}

final class Alg {
	private Alg() {
	}

	static <A extends Comparable<? super A>> A gcd(A a, A b, EuclideanRing<A> ring) {
		AbelianGroup<A> add = ring.add();
		A zero = add.e();
		int sa = a.compareTo(zero);
		int sb = b.compareTo(zero);
		if (sa == 0)
			return b;
		if (sb == 0)
			return a;
		if (sa < 0)
			a = add.inv(a);
		if (sb < 0)
			b = add.inv(b);
		if (a.compareTo(b) < 0) {
			A tmp = a;
			a = b;
			b = tmp;
		}
		while (true) {
			A c = ring.mod(a, b);
			if (c.compareTo(zero) == 0)
				return b;
			a = b;
			b = c;
		}
	}

	static long gcd(long a, long b) {
		if (a == 0)
			return b;
		if (b == 0)
			return a;
		if (a < 0)
			a = -a;
		if (b < 0)
			b = -b;
		if (a < b) {
			a ^= b;
			b ^= a;
			a ^= b;
		}
		while (true) {
			long c = a % b;
			if (c == 0)
				return b;
			a = b;
			b = c;
		}
	}

	static int gcd(int a, int b) {
		return (int) gcd((long) a, (long) b);
	}

	static <A extends Comparable<A>> int[] lis(F.IX<A> access, int size) {
		Object[] dp = new Object[size];
		int[][] dpIndices = new int[size][2];
		dp[0] = access.f(0);
		int len = 1;
		int lidx = 0;
		for (int i = 1; i < size; i++) {
			A ai = access.f(i);
			@SuppressWarnings("unchecked")
			int idx = U.searchI(-1, len, j -> ai.compareTo((A) dp[j]) <= 0); // replace <= with < to return NLDS
			dp[idx] = ai;
			dpIndices[idx][0] = i;
			if (idx == len) {
				lidx = i;
				len++;
			}
			if (idx > 0)
				dpIndices[i][1] = dpIndices[idx - 1][0];
		}
		int[] res = new int[len];
		res[len - 1] = lidx;
		for (int i = len - 1; i >= 0; i--) {
			res[i] = lidx;
			lidx = dpIndices[lidx][1];
		}
		return res;
	}

	static <A> A pow(A a, long b, Monoid<A> monoid) {
		A res = monoid.e();
		while (b > 0) {
			if ((b & 1) != 0)
				res = monoid.g(res, a);
			a = monoid.g(a, a);
			b >>= 1;
		}
		return res;
	}

	static <A> A pow(A a, BigInteger b, Monoid<A> monoid) {
		A res = monoid.e();
		while (b.compareTo(BigInteger.ZERO) > 0) {
			if ((b.and(BigInteger.ONE)).intValueExact() != 0)
				res = monoid.g(res, a);
			a = monoid.g(a, a);
			b = b.divide(BigInteger.valueOf(2));
		}
		return res;
	}

	static long pow(long a, long b) {
		long res = 1;
		while (b > 0) {
			if ((b & 1) != 0)
				res *= a;
			a *= a;
			b >>= 1;
		}
		return res;
	}

	static <A extends Comparable<? super A>> T<A, A, A> extgcd(A a, A b, EuclideanRing<A> ring) { // returns (x, y, d) s.t. ax + by = d
		AbelianGroup<A> add = ring.add();
		CommutativeMonoid<A> mul = ring.mul();
		A zero = add.e();
		A one = mul.e();
		A sa = a.compareTo(zero) < 0 ? add.inv(one) : one;
		A sb = b.compareTo(zero) < 0 ? add.inv(one) : one;
		a = mul.g(a, sa);
		b = mul.g(b, sb);
		A x = one;
		A y = zero;
		A z = zero;
		A w = one;
		while (b.compareTo(zero) > 0) {
			A q = ring.div(a, b);
			A t = z;
			z = add.g(x, add.inv(mul.g(q, z)));
			x = t;
			t = w;
			w = add.g(y, add.inv(mul.g(q, w)));
			y = t;
			t = b;
			b = add.g(a, add.inv(mul.g(q, b)));
			a = t;
		}
		return T.make(mul.g(x, sa), mul.g(y, sb), a);
	}

	static TL extgcd(long a, long b) {
		int sa = a < 0 ? -1 : 1;
		int sb = b < 0 ? -1 : 1;
		a *= sa;
		b *= sb;
		long x = 1;
		long y = 0;
		long z = 0;
		long w = 1;
		while (b > 0) {
			long q = a / b;
			long t = z;
			z = x - q * z;
			x = t;
			t = w;
			w = y - q * w;
			y = t;
			t = b;
			b = a - q * b;
			a = t;
		}
		return TL.make(x * sa, y * sb, a);
	}

	static TI extgcd(int a, int b) {
		return extgcd((long) a, (long) b).toInt();
	}

	static ArrayList<PI> factorize(int n) { // factor, exponent
		ArrayList<PI> res = new ArrayList<PI>();
		for (int i = 2; i * i <= n; i++) {
			int count = 0;
			while (n % i == 0) {
				n /= i;
				count++;
			}
			if (count > 0)
				res.add(PI.make(i, count));
		}
		if (n > 1)
			res.add(PI.make(n, 1));
		return res;
	}

	static ArrayList<PL> factorize(long n) { // factor, exponent
		ArrayList<PL> res = new ArrayList<PL>();
		for (int i = 2; i * i <= n; i++) {
			int count = 0;
			while (n % i == 0) {
				n /= i;
				count++;
			}
			if (count > 0)
				res.add(PL.make(i, count));
		}
		if (n > 1)
			res.add(PL.make(n, 1));
		return res;
	}

	static <A> A arithSum(A a, A d, long num, Ring<A> ring) {
		return arithGeomSum(a, d, ring.one(), ring.one(), num, ring);
	}

	static <A> A geomSum(A b, A r, long num, Ring<A> ring) {
		return arithGeomSum(ring.one(), ring.zero(), b, r, num, ring);
	}

	static <A> A arithGeomSum(A a, A d, A b, A r, long num, Ring<A> ring) { // Σ(a+(i-1)d)br^(i-1)
		AbelianGroup<A> add = ring.add();
		Monoid<A> mul = ring.mul();
		Monoid<MN<A>> matMul = Mat.mulRing(3, ring);
		A zero = ring.zero();
		A one = ring.one();
		MN<A> mat = pow(Mat.make(new Object[][] { { r, d, a }, { zero, r, r }, { zero, zero, one } }), num - 1, matMul);
		return mul.g(add.g(add.g(mul.g(mat.at(0, 0), a), mul.g(mat.at(0, 1), r)), mat.at(0, 2)), b);
	}
}

class MN<A> {
	final int m;
	final int n;
	private final Object[][] as;

	static class Scalar<A> extends MN<A> {
		final long l;
		A a;

		Scalar(int m, int n, long l, A zero, A oneTimesL) {
			super(m, n, (i, j) -> i == j ? oneTimesL : zero);
			this.a = oneTimesL;
			this.l = l;
		}
	}

	MN(int m, int n, Mat.Accessor<A> accessor) {
		this.m = m;
		this.n = n;
		as = new Object[m][n];
		for (int i = 0; i < m; i++) {
			for (int j = 0; j < n; j++) {
				as[i][j] = accessor.at(i, j);
			}
		}
	}

	public String toString() {
		String s = "";
		for (int i = 0; i < m; i++) {
			s += i == 0 ? "[[ " : " [ ";
			for (int j = 0; j < n; j++) {
				s += (j == 0 ? "" : ", ") + as[i][j];
			}
			s += i == m - 1 ? " ]]" : " ]\n";
		}
		return s;
	}

	@SuppressWarnings("unchecked")
	A at(int i, int j) {
		return (A) as[i][j];
	}

	@SuppressWarnings("unchecked")
	A[][] toArray() {
		A[][] res = (A[][]) Array.newInstance(as[0][0].getClass(), m, n);
		for (int i = 0; i < m; i++) {
			for (int j = 0; j < n; j++) {
				res[i][j] = (A) as[i][j];
			}
		}
		return res;
	}

	int[][] toIntArray() {
		int[][] res = new int[m][n];
		for (int i = 0; i < m; i++) {
			for (int j = 0; j < n; j++) {
				res[i][j] = (int) as[i][j];
			}
		}
		return res;
	}

	long[][] toLongArray() {
		long[][] res = new long[m][n];
		for (int i = 0; i < m; i++) {
			for (int j = 0; j < n; j++) {
				res[i][j] = (long) as[i][j];
			}
		}
		return res;
	}

	double[][] toDoubleArray() {
		double[][] res = new double[m][n];
		for (int i = 0; i < m; i++) {
			for (int j = 0; j < n; j++) {
				res[i][j] = (double) as[i][j];
			}
		}
		return res;
	}
}

class Mat<A> {
	interface Accessor<A> {
		A at(int i, int j);
	}

	private static <A> MN.Scalar<A> make(int m, int n, long l, A zero, A oneTimesL) {
		return new MN.Scalar<A>(m, n, l, zero, oneTimesL);
	}

	static <A> MN<A> make(int m, int n, Accessor<A> accessor) {
		return new MN<A>(m, n, accessor);
	}

	static <A> MN<A> make(int m, int n, A[][] as) {
		return new MN<A>(m, n, (i, j) -> as[i][j]);
	}

	static MN<Long> make(int[][] as) {
		return new MN<Long>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> (long) as[i][j]);
	}

	static MN<Long> make(long[][] as) {
		return new MN<Long>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> as[i][j]);
	}

	static MN<Double> make(double[][] as) {
		return new MN<Double>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> as[i][j]);
	}

	static <A> MN<A> make(int[][] as, F.IX<A> toA) {
		return new MN<A>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> toA.f(as[i][j]));
	}

	static <A> MN<A> make(long[][] as, F.LX<A> toA) {
		return new MN<A>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> toA.f(as[i][j]));
	}

	static <A> MN<A> make(double[][] as, F.DX<A> toA) {
		return new MN<A>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> toA.f(as[i][j]));
	}

	static <A, B> MN<A> make(B[][] as, F.XX<B, A> toA) {
		return new MN<A>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> toA.f(as[i][j]));
	}

	@SuppressWarnings("unchecked")
	static <A> MN<A> make(Object[][] as) {
		return new MN<A>(as.length, as.length == 0 ? 0 : as[0].length, (i, j) -> (A) as[i][j]);
	}

	static <A> MN<A> eye(int n, Ring<A> ring) {
		return make(n, n, (i, j) -> i == j ? ring.mul().e() : ring.add().e());
	}

	static <A> AbelianGroup<MN<A>> add(int m, int n, Ring<A> ring) {
		return AbelianGroup.make((a, b) -> add(a, b, ring), a -> neg(a, ring), make(m, n, 0, ring.zero(), ring.zero()));
	}

	static <A> Monoid<MN<A>> mulRing(int n, Ring<A> ring) {
		return Monoid.make((a, b) -> mul(a, b, ring), make(n, n, 1, ring.zero(), ring.one()));
	}

	static <A> AbelianGroup<MN<A>> mulField(int n, Field<A> field) {
		return AbelianGroup.make((a, b) -> mul(a, b, field), a -> inv(a, field),
				make(n, n, 1, field.zero(), field.one()));
	}

	static <A> MN<A> add(MN<A> a, MN<A> b, Ring<A> ring) {
		int m = U.max(a.m, b.m);
		int n = U.max(a.n, b.n);
		AbelianGroup<A> add = ring.add();
		if (a instanceof MN.Scalar && b instanceof MN.Scalar) {
			MN.Scalar<A> as = ((MN.Scalar<A>) a);
			MN.Scalar<A> bs = ((MN.Scalar<A>) b);
			return make(m, n, as.l + bs.l, ring.zero(), add.g(as.a, bs.a));
		}
		return make(m, n, (i, j) -> add.g(a.at(i, j), b.at(i, j)));
	}

	static <A> MN<A> neg(MN<A> a, Ring<A> ring) {
		if (a instanceof MN.Scalar) {
			MN.Scalar<A> as = ((MN.Scalar<A>) a);
			return make(a.m, a.n, -as.l, ring.zero(), ring.add().inv(as.a));
		}
		return make(a.m, a.n, (i, j) -> ring.add().inv(a.at(i, j)));
	}

	static <A> MN<A> mul(MN<A> a, MN<A> b, Ring<A> ring) {
		int m = a.m;
		int u = U.max(a.n, b.m);
		int n = b.n;
		AbelianGroup<A> add = ring.add();
		Monoid<A> mul = ring.mul();
		if (a instanceof MN.Scalar && b instanceof MN.Scalar) {
			MN.Scalar<A> as = ((MN.Scalar<A>) a);
			MN.Scalar<A> bs = ((MN.Scalar<A>) b);
			return make(m, n, as.l * bs.l, ring.zero(), mul.g(as.a, bs.a));
		}
		return make(m, n, (i, j) -> {
			A res = ring.zero();
			for (int k = 0; k < u; k++)
				res = add.g(res, mul.g(a.at(i, k), b.at(k, j)));
			return res;
		});
	}

	static <A> A det(MN<A> a, Field<A> field) {
		return detInv(a, field).a;
	}

	static <A> MN<A> inv(MN<A> a, Field<A> field) {
		UP<A, MN<A>> detInv = detInv(a, field);
		if (detInv.a.equals(field.zero()))
			throw new ArithmeticException("inverse does not exist: det=0");
		return detInv.b;
	}

	@SuppressWarnings("unchecked")
	private static <A> UP<A, MN<A>> detInv(MN<A> a, Field<A> field) {
		if (a.m != a.n)
			throw new IllegalArgumentException("matrix not square");
		if (field.zero() instanceof Long) {
			UP<Long, MN<Long>> detInv = detInvLong((MN<Long>) a, (Field<Long>) field);
			return UP.make((A) detInv.a, (MN<A>) detInv.b);
		}
		int n = a.n;
		AbelianGroup<A> add = field.add();
		AbelianGroup<A> mul = field.mul();
		A zero = field.zero();
		A one = field.one();
		A[][] m1 = a.toArray();
		A[][] m2 = eye(n, field).toArray();
		A res = one;
		int sign = 1;
		for (int i = 0; i < n; i++) {
			int pivot = -1;
			for (int j = i; j < n; j++) {
				if (!m1[j][i].equals(zero)) {
					pivot = j;
					break;
				}
			}
			if (pivot == -1) {
				return UP.make(zero, null);
			}
			if (pivot != i) {
				sign = -sign;
				A tmp;
				for (int j = i; j < n; j++) { // [0, i) are zero
					tmp = m1[i][j];
					m1[i][j] = m1[pivot][j];
					m1[pivot][j] = tmp;
				}
				for (int j = 0; j < n; j++) {
					tmp = m2[i][j];
					m2[i][j] = m2[pivot][j];
					m2[pivot][j] = tmp;
				}
			}
			A d = m1[i][i];
			res = mul.g(res, d);
			d = mul.inv(d);
			m1[i][i] = one;
			for (int j = i + 1; j < n; j++) {
				m1[i][j] = mul.g(m1[i][j], d);
			}
			for (int j = 0; j < n; j++) {
				m2[i][j] = mul.g(m2[i][j], d);
			}
			for (int j = 0; j < n; j++) {
				if (i == j)
					continue;
				A mult = m1[j][i];
				m1[j][i] = zero;
				for (int k = i + 1; k < n; k++) {
					m1[j][k] = add.g(m1[j][k], add.inv(mul.g(m1[i][k], mult)));
				}
				for (int k = 0; k < n; k++) {
					m2[j][k] = add.g(m2[j][k], add.inv(mul.g(m2[i][k], mult)));
				}
			}
		}
		return UP.make(sign == 1 ? res : add.inv(res), make(m2));
	}

	private static UP<Long, MN<Long>> detInvLong(MN<Long> a, Field<Long> field) {
		if (a.m != a.n)
			throw new IllegalArgumentException("matrix not square");
		int n = a.n;
		AbelianGroup<Long> add = field.add();
		AbelianGroup<Long> mul = field.mul();
		long zero = field.zero();
		long one = field.one();
		long[][] m1 = a.toLongArray();
		long[][] m2 = eye(n, field).toLongArray();
		long res = one;
		int sign = 1;
		for (int i = 0; i < n; i++) {
			int pivot = -1;
			for (int j = i; j < n; j++) {
				if (m1[j][i] != zero) {
					pivot = j;
					break;
				}
			}
			if (pivot == -1) {
				return UP.make(zero, null);
			}
			if (pivot != i) {
				sign = -sign;
				long tmp;
				for (int j = i; j < n; j++) { // [0, i) are zero
					tmp = m1[i][j];
					m1[i][j] = m1[pivot][j];
					m1[pivot][j] = tmp;
				}
				for (int j = 0; j < n; j++) {
					tmp = m2[i][j];
					m2[i][j] = m2[pivot][j];
					m2[pivot][j] = tmp;
				}
			}
			long d = m1[i][i];
			res = mul.g(res, d);
			d = mul.inv(d);
			m1[i][i] = one;
			for (int j = i + 1; j < n; j++) {
				m1[i][j] = mul.g(m1[i][j], d);
			}
			for (int j = 0; j < n; j++) {
				m2[i][j] = mul.g(m2[i][j], d);
			}
			for (int j = 0; j < n; j++) {
				if (i == j)
					continue;
				long mult = m1[j][i];
				m1[j][i] = zero;
				for (int k = i + 1; k < n; k++) {
					m1[j][k] = add.g(m1[j][k], add.inv(mul.g(m1[i][k], mult)));
				}
				for (int k = 0; k < n; k++) {
					m2[j][k] = add.g(m2[j][k], add.inv(mul.g(m2[i][k], mult)));
				}
			}
		}
		return UP.make(sign == 1 ? res : add.inv(res), make(m2));
	}

	static <A> Ring<MN<A>> ring(int n, Ring<A> ring) {
		return Ring.make(add(n, n, ring), mulRing(n, ring));
	}

	static <A> Field<MN<A>> field(int n, Field<A> field) {
		return Field.make(add(n, n, field), mulField(n, field));
	}

	static <A> VM<MN<A>> vm(int n, Ring<A> ring) {
		if (ring instanceof Field)
			return vmField(n, (Field<A>) ring);
		return vmRing(n, ring);
	}

	private static <A> VM<MN<A>> vmRing(int n, Ring<A> ring) {
		Ring<MN<A>> matRing = ring(n, ring);
		Monoid<MN<A>> matMul = matRing.mul();
		AbelianGroup<A> add = ring.add();
		Monoid<A> mul = ring.mul();
		A zero = add.e();
		A one = mul.e();
		return new VM<MN<A>>(matRing, null, null, (a, b) -> {
			if (b instanceof MN.Scalar) {
				MN.Scalar<A> bs = ((MN.Scalar<A>) b);
				if (bs.l < 0)
					throw new RuntimeException("pow(MN, <0) is not defined");
				return Alg.pow(a, bs.l, matMul);
			}
			throw new RuntimeException("pow(MN, MN) is not defined");
		}, null, l -> make(n, n, l, zero, Alg.pow(one, l, add)), a -> a);
	}

	private static <A> VM<MN<A>> vmField(int n, Field<A> field) {
		Field<MN<A>> matField = field(n, field);
		AbelianGroup<MN<A>> matMul = matField.mul();
		A zero = field.zero();
		A one = field.one();
		return new VM<MN<A>>(matField, (a, b) -> {
			if (b instanceof MN.Scalar) {
				MN.Scalar<A> bs = ((MN.Scalar<A>) b);
				if (bs.l < 0)
					return Alg.pow(inv(a, field), -bs.l, matMul);
				return Alg.pow(a, bs.l, matMul);
			}
			throw new RuntimeException("pow(MN, MN) is not defined");
		}, a -> inv(a, field), l -> make(n, n, l, zero, Alg.pow(one, l, field.add())), a -> a);
	}
}

class VM<A> {
	private final HashMap<String, A> env;
	private final Evaluator<A> etor;
	private final F.XX<A, A> filter;

	VM(F.XXX<A, A, A> add, F.XXX<A, A, A> sub, F.XXX<A, A, A> mul, F.XXX<A, A, A> div, F.XXX<A, A, A> mod,
			F.XXX<A, A, A> pow, F.XX<A, A> neg, F.XX<A, A> fact, F.LX<A> fromInt, F.XX<A, A> filter) {
		env = new HashMap<String, A>();
		etor = SimpleLang.makeEvaluator((s, a) -> {
			env.put(s, a);
			return a;
		}, add, sub, mul, div, mod, pow, neg, fact, fromInt, s -> {
			if (env.containsKey(s))
				return env.get(s);
			throw new RuntimeException("no such variable: " + s);
		});
		this.filter = filter;
	}

	VM(Ring<A> ring, F.XXX<A, A, A> div, F.XXX<A, A, A> mod, F.XXX<A, A, A> pow, F.XX<A, A> fact, F.LX<A> fromInt,
			F.XX<A, A> filter) {
		this(ring.add()::g, (a, b) -> ring.add().g(a, ring.add().inv(b)), ring.mul()::g, div, mod, pow, ring.add()::inv,
				fact, fromInt, filter);
	}

	VM(EuclideanRing<A> ring, F.XXX<A, A, A> pow, F.XX<A, A> fact, F.LX<A> fromInt, F.XX<A, A> filter) {
		this(ring, ring::div, ring::mod, pow, fact, fromInt, filter);
	}

	void clear() {
		env.clear();
	}

	A get(String id) {
		return env.get(id);
	}

	void print(String id, F.XV<String> printer) {
		printer.f("" + get(id));
	}

	@SafeVarargs
	final void set(String idsSp, A... as) {
		String[] ids = idsSp.trim().split(" +");
		int n = ids.length;
		if (as.length != n)
			throw new IllegalArgumentException("argument size mismatch: " + n + " != " + as.length);
		for (int i = 0; i < n; i++)
			set(ids[i], as[i]);
	}

	void set(String id, A a) {
		env.put(id, filter.f(a));
	}

	A run(String expr) {
		return AST.eval(SimpleLang.parse(expr), etor);
	}
}

class AST { // Abstract Syntax Tree
	static class AssignOp extends AST {
		String id;
		AST r;

		AssignOp(String id, AST r) {
			this.id = id;
			this.r = r;
		}

		public String toString() {
			return "Assign(" + id + ", " + r + ")";
		}
	}

	static class Multi extends AST {
		ArrayList<AST> as;

		Multi(ArrayList<AST> as) {
			this.as = as;
		}

		public String toString() {
			String s = "";
			for (AST a : as)
				s += s.isEmpty() ? a : "; " + a;
			return "Multi(" + s + ")";
		}
	}

	static class BinOp extends AST {
		AST l;
		AST r;
		String op;

		BinOp(AST l, String op, AST r) {
			this.l = l;
			this.op = op;
			this.r = r;
		}

		public String toString() {
			return "BinOp(" + l + ", " + op + ", " + r + ")";
		}
	}

	static class UnOp extends AST {
		AST a;
		String op;

		UnOp(String op, AST a) {
			this.op = op;
			this.a = a;
		}

		public String toString() {
			return "UnOp(" + op + ", " + a + ")";
		}
	}

	static class Int extends AST {
		long v;

		Int(long v) {
			this.v = v;
		}

		public String toString() {
			return "Int(" + v + ")";
		}
	}

	static class Id extends AST {
		String s;

		Id(String s) {
			this.s = s;
		}

		public String toString() {
			return "Id(" + s + ")";
		}
	}

	static <A> A eval(AST a, Evaluator<A> etor) {
		if (a instanceof AssignOp)
			return etor.assign(((AssignOp) a).id, eval(((AssignOp) a).r, etor));
		if (a instanceof Multi) {
			A last = null;
			for (AST a2 : ((Multi) a).as)
				last = eval(a2, etor);
			return last;
		}
		if (a instanceof BinOp)
			return etor.binOp(((BinOp) a).op, eval(((BinOp) a).l, etor), eval(((BinOp) a).r, etor));
		if (a instanceof UnOp)
			return etor.unOp(((UnOp) a).op, eval(((UnOp) a).a, etor));
		if (a instanceof Int)
			return etor.fromInt(((Int) a).v);
		if (a instanceof Id)
			return etor.id(((Id) a).s);
		throw new RuntimeException("unexpected ast: " + a);
	}
}

interface Evaluator<A> {
	A assign(String s, A a);

	A binOp(String op, A a, A b);

	A unOp(String op, A a);

	A fromInt(long a);

	A id(String s);
}

class Seq<A> {
	ArrayList<A> as;
	int ptr;

	Seq(ArrayList<A> as) {
		this.as = as;
		ptr = 0;
	}

	boolean hasNext(int num) {
		return ptr + num < as.size();
	}

	boolean hasNext() {
		return hasNext(0);
	}

	A next(int num) {
		return ptr + num < as.size() ? as.get(ptr + num) : null;
	}

	A next() {
		return next(0);
	}

	A read() {
		return hasNext() ? as.get(ptr++) : null;
	}

	A read(A a) {
		if (!hasNext())
			throw new RuntimeException("unexpected EOF");
		if (!isNext(a))
			throw new RuntimeException("expected " + a + " but got" + next());
		return read();
	}

	A read(F.XX<A, Boolean> f) {
		if (!hasNext())
			throw new RuntimeException("unexpected EOF");
		if (!f.f(next()))
			throw new RuntimeException("f(" + next() + ") returned false");
		return read();
	}

	boolean isNext(A a) {
		return a.equals(next());
	}

	boolean isNext(@SuppressWarnings("unchecked") A... as) {
		for (A a : as)
			if (isNext(a))
				return true;
		return false;
	}

	boolean isNext(F.XX<A, Boolean> f) {
		return hasNext() && f.f(next());
	}

	A readIf(F.XX<A, Boolean> f) {
		if (isNext(f))
			return read();
		return null;
	}

	A readIf(@SuppressWarnings("unchecked") A... as) {
		for (A a : as)
			if (isNext(a)) {
				ptr++;
				return a;
			}
		return null;
	}

	public String toString() {
		return as.toString();
	}
}

class Token extends P<String, Integer> {
	private static final int ID = 1;
	private static final int SYM = 2;
	private static final int INT = 3;

	Token(String s, int type) {
		super(s, type);
	}

	boolean is(String s) {
		return a.equals(s);
	}

	static Token ofId(String s) {
		return new Token(s, ID);
	}

	static Token ofSym(String s) {
		return new Token(s, SYM);
	}

	static Token ofInt(String s) {
		return new Token(s, INT);
	}

	boolean isId() {
		return b == ID;
	}

	boolean isSym() {
		return b == SYM;
	}

	boolean isInt() {
		return b == INT;
	}
}

class SimpleLang {
	private static class Tokenizer {
		private static final String SYMS = "+-*/%^!()=;";

		Seq<Character> sc;
		ArrayList<Token> ts;

		Tokenizer(String s) {
			sc = new Seq<Character>(U.make(s.length(), i -> s.charAt(i)));
		}

		Seq<Token> parse() {
			ts = new ArrayList<Token>();
			parseAll();
			return new Seq<Token>(ts);
		}

		private static boolean isSpace(char c) {
			return c == ' ' || c == '\t' || c == '\r' || c == '\n';
		}

		private static boolean isDigit(char c) {
			return c >= '0' && c <= '9';
		}

		private static boolean isSymbol(char c) {
			return SYMS.indexOf(c) != -1;
		}

		private static boolean isAlpha(char c) {
			return c >= 'a' && c <= 'z' || c >= 'A' && c <= 'Z' || c == '_';
		}

		private void parseAll() {
			while (sc.hasNext()) {
				while (sc.isNext(Tokenizer::isSpace))
					sc.read();
				if (!sc.hasNext())
					break;
				if (sc.isNext(Tokenizer::isAlpha)) {
					parseId();
				} else if (sc.isNext(Tokenizer::isDigit)) {
					parseInt();
				} else if (sc.isNext(Tokenizer::isSymbol)) {
					parseSym();
				} else {
					throw new RuntimeException("invalid character: " + sc.next());
				}
			}
		}

		private void parseId() {
			String s = sc.read().toString();
			while (sc.isNext(Tokenizer::isAlpha) || sc.isNext(Tokenizer::isDigit))
				s += sc.read().toString();
			ts.add(Token.ofId(s));
		}

		private void parseInt() {
			String s = sc.read().toString();
			while (sc.isNext(Tokenizer::isDigit))
				s += sc.read().toString();
			ts.add(Token.ofInt(s));
		}

		private void parseSym() {
			String s = sc.read().toString();
			ts.add(Token.ofSym(s));
		}
	}

	private static class Parser {
		Seq<Token> ts;

		Parser(Seq<Token> ts) {
			this.ts = ts;
		}

		AST parse() {
			return parseExpr();
		}

		private AST parseExpr() {
			return parseMultiExpr();
		}

		private AST parseMultiExpr() {
			readSemicolons();
			AST a = parseAssignExpr();
			if (readSemicolons()) {
				ArrayList<AST> as = new ArrayList<AST>();
				as.add(a);
				do {
					if (!ts.hasNext())
						break;
					as.add(parseAssignExpr());
				} while (readSemicolons());
				if (as.size() == 1)
					return a;
				return new AST.Multi(as);
			}
			return a;
		}

		private boolean readSemicolons() {
			if (!ts.isNext(t -> t.is(";")))
				return false;
			do {
				ts.read();
			} while (ts.isNext(t -> t.is(";")));
			return true;
		}

		private AST parseAssignExpr() {
			AST a = parseAddSubOp();
			if (ts.isNext(t -> t.is("="))) {
				ts.read();
				if (!(a instanceof AST.Id))
					throw new RuntimeException("cannot assign to " + a);
				return new AST.AssignOp(((AST.Id) a).s, parseAssignExpr());
			}
			return a;
		}

		private AST parseAddSubOp() {
			AST a = parseMulDivModOp();
			while (ts.isNext(t -> t.is("+") || t.is("-")))
				a = new AST.BinOp(a, ts.read().a, parseMulDivModOp());
			return a;
		}

		private AST parseMulDivModOp() {
			AST a = parsePowOp();
			while (ts.isNext(t -> t.is("*") || t.is("/") || t.is("%")))
				a = new AST.BinOp(a, ts.read().a, parsePowOp());
			return a;
		}

		private AST parsePowOp() {
			AST a = parseNegateOp();
			if (ts.isNext(t -> t.is("^")))
				return new AST.BinOp(a, ts.read().a, parsePowOp());
			return a;
		}

		private AST parseNegateOp() {
			if (ts.isNext(t -> t.is("-")))
				return new AST.UnOp(ts.read().a, parseNegateOp());
			return parseFactOp();
		}

		private AST parseFactOp() {
			AST a = parsePrimary();
			while (ts.isNext(t -> t.is("!")))
				a = new AST.UnOp(ts.read().a, a);
			return a;
		}

		private AST parsePrimary() {
			if (ts.isNext(t -> t.isId()))
				return new AST.Id(ts.read().a);
			if (ts.isNext(t -> t.isInt()))
				return new AST.Int(Long.parseLong(ts.read().a));
			if (ts.readIf(t -> t.is("(")) != null) {
				AST e = parseExpr();
				ts.read(t -> t.is(")"));
				return e;
			}
			throw new RuntimeException("unexpected token: " + ts.next());
		}
	}

	static HashMap<String, AST> cache = new HashMap<String, AST>();

	static <A> Evaluator<A> makeEvaluator(F.XXX<String, A, A> assign, F.XXX<A, A, A> add, F.XXX<A, A, A> sub,
			F.XXX<A, A, A> mul, F.XXX<A, A, A> div, F.XXX<A, A, A> mod, F.XXX<A, A, A> pow, F.XX<A, A> neg,
			F.XX<A, A> fact, F.LX<A> fromInt, F.XX<String, A> id) {
		return new Evaluator<A>() {
			public A assign(String s, A a) {
				return assign.f(s, a);
			}

			public A binOp(String op, A a, A b) {
				switch (op) {
				case "+":
					return add.f(a, b);
				case "-":
					return sub.f(a, b);
				case "*":
					return mul.f(a, b);
				case "/":
					return div.f(a, b);
				case "%":
					return mod.f(a, b);
				case "^":
					return pow.f(a, b);
				}
				throw new RuntimeException("invalid binOp: " + op);
			}

			public A unOp(String op, A a) {
				switch (op) {
				case "-":
					return neg.f(a);
				case "!":
					return fact.f(a);
				}
				throw new RuntimeException("invalid unOp: " + op);
			}

			public A fromInt(long a) {
				return fromInt.f(a);
			}

			public A id(String s) {
				return id.f(s);
			}
		};
	}

	static AST parse(String s) {
		if (cache.containsKey(cache)) {
			return cache.get(s);
		}
		Tokenizer l = new Tokenizer(s);
		Seq<Token> ts = l.parse();
		Parser p = new Parser(ts);
		AST a = p.parse();
		cache.put(s, a);
		return a;
	}
}

class Mod {
	final long mod;
	final AbelianGroup<Long> add;
	final AbelianGroup<Long> mul;
	final Field<Long> field;
	final VM<Long> vm;
	private final boolean prime;
	private final HashMap<Long, Integer> logMap;
	private long[] facts;
	private long[] invs;
	private long[] invFacts;
	private long[] factors;

	Mod(long mod) {
		this.mod = mod;
		prepareFacts(0);
		prime = BigInteger.valueOf(mod).isProbablePrime(100);
		add = AbelianGroup.make((a, b) -> (a + b) % mod, a -> mod - a, 0l);
		mul = AbelianGroup.make((a, b) -> (a * b) % mod, a -> {
			if (prime)
				return pow(a, mod - 2);
			TL t = Alg.extgcd(a, mod);
			if (t.c != 1)
				throw new ArithmeticException("inv(" + a + ") does not exist");
			return (t.a % mod + mod) % mod;
		}, 1l);
		field = Field.make(add, mul);
		logMap = new HashMap<Long, Integer>();
		vm = new VM<Long>(field, this::pow, this::fact, a -> (a % mod + mod) % mod, a -> (a % mod + mod) % mod);
	}

	long fact(long a) {
		if (a >= Integer.MAX_VALUE)
			throw new RuntimeException("fact(" + a + ") too big");
		prepareFacts((int) a);
		return facts[(int) (a % mod)];
	}

	long invFact(int a) {
		prepareFacts(a);
		return invFacts[(int) (a % mod)];
	}

	long inv(long a) {
		return mul.inv(a);
	}

	long pow(long a, long b) {
		if (b == 0)
			return 1;
		if (b == 1)
			return a;
		if (b < 0) {
			a = inv(a);
			b = -b;
		}
		a %= mod;
		long res = 1;
		while (b > 0) {
			if ((b & 1) != 0)
				res = res * a % mod;
			a = a * a % mod;
			b >>= 1;
		}
		return res;
	}

	long order(long a) { // computes the order of `a` in O(sqrt(mod)) time
		a %= mod;
		if (a == 0)
			return 0;
		if (a == 1)
			return 1;
		if (factors == null) {
			ArrayList<Long> fs = new ArrayList<Long>();
			for (long i = 2; i * i < mod; i++) {
				if ((mod - 1) % i == 0)
					fs.add(i);
			}
			factors = new long[fs.size()];
			for (int i = 0; i < fs.size(); i++) {
				factors[i] = fs.get(i);
			}
		}
		for (long f : factors) {
			if (pow(a, f) == 1)
				return f;
		}
		return mod - 1;
	}

	PL log(long a, long b) { // log_b(a) in O(sqrt(mod)) time
		a %= mod;
		b %= mod;
		if (b == 1 || b == 0)
			return a == b ? PL.make(1, 1) : PL.make(-1, 0);
		if (a == 0)
			return PL.make(-1, 0);
		long order = order(b);
		if (a == 1)
			return PL.make(0, order);
		long orderSqrtL = sqrtCeil(order);
		if (orderSqrtL > Integer.MAX_VALUE)
			throw new RuntimeException("order(" + b + ") too big: " + order);
		int orderSqrt = (int) sqrtCeil(order);
		logMap.clear();
		logMap.put(1l, 0);
		long p = 1;
		for (int i = 1; i < orderSqrt; i++) {
			p = p * b % mod;
			logMap.put(p, i);
		}
		long ib = pow(b, mod - orderSqrt - 1);
		p = a;
		for (int i = 1; i < orderSqrt; i++) {
			p = p * ib % mod;
			if (logMap.containsKey(p))
				return PL.make((i * orderSqrt + logMap.get(p)) % order, order);
		}
		return PL.make(-1, 0);
	}

	private long sqrtCeil(long a) {
		return U.searchL((long) Math.sqrt(a * 0.9), (long) Math.sqrt(a * 1.1) + 1, mid -> mid * mid >= a);
	}

	private void prepareFacts(int n) {
		if (facts == null) {
			facts = new long[1024];
			invs = new long[1024];
			invFacts = new long[1024];
			prepareFactsIn(0, 1024);
		}
		if (n >= mod)
			n = (int) (mod - 1);
		while (facts.length <= n) {
			int prevL = facts.length;
			int newL = prevL << 1;
			facts = Arrays.copyOf(facts, newL);
			invs = Arrays.copyOf(invs, newL);
			invFacts = Arrays.copyOf(invFacts, newL);
			prepareFactsIn(prevL, newL);
		}
	}

	private void prepareFactsIn(int from, int until) {
		if (until > mod)
			until = (int) mod;
		for (int i = from; i < until; i++) {
			if (i == 0) {
				facts[0] = 1;
				invs[0] = 0;
				invFacts[0] = 1;
				continue;
			}
			if (i == 1) {
				facts[1] = 1;
				invs[1] = 1;
				invFacts[1] = 1;
				continue;
			}
			facts[i] = facts[i - 1] * i % mod;
			invs[i] = (mod - mod / i) * invs[(int) (mod % i)] % mod;
			invFacts[i] = invFacts[i - 1] * invs[i] % mod;
		}
	}
}