Problem: Japate, while traveling through the forest of Mala, saw N bags of gold lying in a row. Each bag has some distinct weight of gold between 1 to N. Japate can carry only one bag of gold with him, so he uses the following strategy to choose a bag.

Initially, he starts with an empty bag (zero weight). He considers the bags in some order. If the current bag has a higher weight than the bag in his hand, he picks the current bag.

Japate put the bags in some order. Japate realizes that he will pick A bags, if he starts picking bags from the front, and will pick B bags, if he starts picking bags from the back. By picking we mean replacing the bag in his hand with the current one.

Now he wonders how many permutations of bags are possible, in which he picks A bags from the front and B bags from back using the above strategy.

Since the answer can be very large, output it modulo 998244353.

Input

The only line of input contains three space separated integers N (1 ≤ N ≤ 105), A and B (0 ≤ A, B ≤ N).

Output

Output a single integer — the number of valid permutations modulo 998244353.

Examples

Input

1 1 1


Output

1

Input

2 1 1


Output

0

Input

2 2 1


Output

1

Input

5 2 2


Output

22

Note

In sample case 1, the only possible permutation is [1]

In sample cases 2 and 3, only two permutations of size 2 are possible:{[1, 2], [2, 1]}. The values of a and b for first permutation is 2 and 1, and for the second permutation these values are 1 and 2. 

In sample case 4, out of 120 permutations of [1, 2, 3, 4, 5] possible, only 22 satisfy the given constraints of a and b.

Solution (in JAVA):

Solution:


import java.io.FileInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.math.BigInteger;
import java.nio.charset.Charset;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;
import java.util.Random;

public class CF960G {
    public static void main(String[] args) throws Exception {
        boolean local = System.getSecurityManager() == null;
        boolean async = false;

        Charset charset = Charset.forName("ascii");

        FastIO io = local ? new FastIO(new FileInputStream("D:\\DATABASE\\TESTCASE\\Code.in"), System.out, charset) : new FastIO(System.in, System.out, charset);
        Task task = new Task(io, new Debug(local));

        if (async) {
            Thread t = new Thread(null, task, "dalt", 1 << 27);
            t.setPriority(Thread.MAX_PRIORITY);
            t.start();
            t.join();
        } else {
            task.run();
        }

        if (local) {
            io.cache.append("\n\n--memory -- \n" + ((Runtime.getRuntime().totalMemory() - Runtime.getRuntime().freeMemory()) >> 20) + "M");
        }

        io.flush();
    }

    public static class Task implements Runnable {
        final FastIO io;
        final Debug debug;
        int inf = (int) 1e8;

        public Task(FastIO io, Debug debug) {
            this.io = io;
            this.debug = debug;
        }

        @Override
        public void run() {
            solve();
        }

        NumberTheory.Modular mod = new NumberTheory.Modular(998244353);
        NumberTheory.Factorial factorial = new NumberTheory.Factorial(100000, mod);
        NumberTheory.Composite composite = new NumberTheory.Composite(factorial, mod);
        NumberTheory.Log2 log2 = new NumberTheory.Log2();

        public void solve() {
            int n = io.readInt();
            int a = io.readInt();
            int b = io.readInt();
            if (a == 0 || b == 0 || a + -1 + b - 1 > n - 1) {
                io.cache.append(0);
                return;
            }
            int[] stirling = getStirling(n - 1);
            int comp = composite.composite(a - 1 + b - 1, a - 1);
            int ans = mod.mul(comp, stirling[a - 1 + b - 1]);
            io.cache.append(ans);
        }

        public int[] getStirling(int n) {
            int proper = log2.ceilLog(n + 1);
            int[][] ans = new int[4][1 << proper];
            calcStirling(ans[0], ans[1], ans[2], ans[3], n);
            return ans[0];
        }

        public void calcStirling(int[] ans, int[] buf, int[] buf2, int[] r, int n) {
            if (n == 0) {
                ans[0] = 1;
                return;
            }
            if (n % 2 == 1) {
                calcStirling(ans, buf, buf2, r, n - 1);
                for (int i = n; i >= 0; i--) {
                    ans[i] = mod.mul(n - 1, ans[i]);
                    if (i >= 1) {
                        ans[i] = mod.plus(ans[i], ans[i - 1]);
                    }
                }
                return;
            }
            int half = n / 2;
            calcStirling(ans, buf, buf2, r, half);
            int proper = log2.ceilLog((half + 1) * 2 - 1);
            NumberTheoryTransform.prepareReverse(r, proper);
            Arrays.fill(buf, 0, 1 << proper, 0);
            Arrays.fill(buf2, 0, 1 << proper, 0);
            int ni = 1;
            for (int i = 0; i <= half; i++) {
                buf[i] = mod.mul(ans[i], factorial.fact[i]);
                buf2[i] = mod.mul(ni, factorial.inv[i]);
                ni = mod.mul(ni, half);
            }
            reverse(buf, 0, half);
            NumberTheoryTransform.dft(r, buf, proper);
            NumberTheoryTransform.dft(r, buf2, proper);
            NumberTheoryTransform.dotMul(buf, buf2, buf, proper);
            NumberTheoryTransform.idft(r, buf, proper);
            reverse(buf, 0, half);
            for (int i = 0; i <= half; i++) {
                buf[i] = mod.mul(buf[i], factorial.inv[i]);
            }
            Arrays.fill(buf, half + 1, 1 << proper, 0);
            NumberTheoryTransform.dft(r, buf, proper);
            NumberTheoryTransform.dft(r, ans, proper);
            NumberTheoryTransform.dotMul(ans, buf, ans, proper);
            NumberTheoryTransform.idft(r, ans, proper);
        }

        public void reverse(int[] data, int l, int r) {
            while (l < r) {
                int tmp = data[l];
                data[l] = data[r];
                data[r] = tmp;
                l++;
                r--;
            }

        }
    }

    public static class FastIO {
        public final StringBuilder cache = new StringBuilder(1 << 13);
        private final InputStream is;
        private final OutputStream os;
        private final Charset charset;
        private StringBuilder defaultStringBuf = new StringBuilder(1 << 13);
        private byte[] buf = new byte[1 << 13];
        private int bufLen;
        private int bufOffset;
        private int next;

        public FastIO(InputStream is, OutputStream os, Charset charset) {
            this.is = is;
            this.os = os;
            this.charset = charset;
        }

        public FastIO(InputStream is, OutputStream os) {
            this(is, os, Charset.forName("ascii"));
        }

        private int read() {
            while (bufLen == bufOffset) {
                bufOffset = 0;
                try {
                    bufLen = is.read(buf);
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
                if (bufLen == -1) {
                    return -1;
                }
            }
            return buf[bufOffset++];
        }

        public void skipBlank() {
            while (next >= 0 && next <= 32) {
                next = read();
            }
        }

        public int readInt() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            int val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

        public long readLong() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            long val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

        public double readDouble() {
            boolean sign = true;
            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+';
                next = read();
            }

            long val = 0;
            while (next >= '0' && next <= '9') {
                val = val * 10 + next - '0';
                next = read();
            }
            if (next != '.') {
                return sign ? val : -val;
            }
            next = read();
            long radix = 1;
            long point = 0;
            while (next >= '0' && next <= '9') {
                point = point * 10 + next - '0';
                radix = radix * 10;
                next = read();
            }
            double result = val + (double) point / radix;
            return sign ? result : -result;
        }

        public String readString(StringBuilder builder) {
            skipBlank();

            while (next > 32) {
                builder.append((char) next);
                next = read();
            }

            return builder.toString();
        }

        public String readString() {
            defaultStringBuf.setLength(0);
            return readString(defaultStringBuf);
        }

        public int readLine(char[] data, int offset) {
            int originalOffset = offset;
            while (next != -1 && next != '\n') {
                data[offset++] = (char) next;
                next = read();
            }
            return offset - originalOffset;
        }

        public int readString(char[] data, int offset) {
            skipBlank();

            int originalOffset = offset;
            while (next > 32) {
                data[offset++] = (char) next;
                next = read();
            }

            return offset - originalOffset;
        }

        public int readString(byte[] data, int offset) {
            skipBlank();

            int originalOffset = offset;
            while (next > 32) {
                data[offset++] = (byte) next;
                next = read();
            }

            return offset - originalOffset;
        }

        public char readChar() {
            skipBlank();
            char c = (char) next;
            next = read();
            return c;
        }

        public void flush() throws IOException {
            os.write(cache.toString().getBytes(charset));
            os.flush();
            cache.setLength(0);
        }

        public boolean hasMore() {
            skipBlank();
            return next != -1;
        }
    }

    public static class Debug {
        private boolean allowDebug;

        public Debug(boolean allowDebug) {
            this.allowDebug = allowDebug;
        }

        public void assertTrue(boolean flag) {
            if (!allowDebug) {
                return;
            }
            if (!flag) {
                fail();
            }
        }

        public void fail() {
            throw new RuntimeException();
        }

        public void assertFalse(boolean flag) {
            if (!allowDebug) {
                return;
            }
            if (flag) {
                fail();
            }
        }

        private void outputName(String name) {
            System.out.print(name + " = ");
        }

        public void debug(String name, int x) {
            if (!allowDebug) {
                return;
            }

            outputName(name);
            System.out.println("" + x);
        }

        public void debug(String name, long x) {
            if (!allowDebug) {
                return;
            }
            outputName(name);
            System.out.println("" + x);
        }

        public void debug(String name, double x) {
            if (!allowDebug) {
                return;
            }
            outputName(name);
            System.out.println("" + x);
        }

        public void debug(String name, int[] x) {
            if (!allowDebug) {
                return;
            }
            outputName(name);
            System.out.println(Arrays.toString(x));
        }

        public void debug(String name, long[] x) {
            if (!allowDebug) {
                return;
            }
            outputName(name);
            System.out.println(Arrays.toString(x));
        }

        public void debug(String name, double[] x) {
            if (!allowDebug) {
                return;
            }
            outputName(name);
            System.out.println(Arrays.toString(x));
        }

        public void debug(String name, Object x) {
            if (!allowDebug) {
                return;
            }
            outputName(name);
            System.out.println("" + x);
        }

        public void debug(String name, Object... x) {
            if (!allowDebug) {
                return;
            }
            outputName(name);
            System.out.println(Arrays.deepToString(x));
        }
    }

    public static class NumberTheory {
        private static final Random RANDOM = new Random();

        public static class ExtLucasFactorial {
            int exp;
            int fact;
            int p;
            int pc;
            Modular modular;
            Power power;
            ExtGCD extGCD = new ExtGCD();
            int[] g;

            /**
             * O(pc)
             *
             * @param p  the prime
             * @param pc p^c
             * @param g  buffer
             */
            public ExtLucasFactorial(int p, int pc, int[] g) {
                this.p = p;
                this.pc = pc;
                this.g = g;
                modular = new Modular(pc);
                power = new Power(modular);
                g[0] = 1;
                g[1] = 1;
                for (int i = 2; i <= pc; i++) {
                    if (i % p == 0) {
                        g[i] = g[i - 1];
                    } else {
                        g[i] = modular.mul(g[i - 1], i);
                    }
                }
            }

            /**
             * return m! (mod pc) without any factor which is multiple of p.
             * <br>
             * O(\log_2^2{m})
             */
            private int fact(long m) {
                fact = 1;
                exp = 0;
                while (m > 1) {
                    exp += m / p;
                    fact = modular.mul(fact, power.pow(g[pc], m / pc));
                    fact = modular.mul(fact, g[(int) (m % pc)]);
                    m /= p;
                }
                return fact;
            }

            /**
             * Find C(m,n), it means choose n elements from a set whose size is m.
             * <br>
             * O(\log_2^2{m})
             */
            public int composite(long m, long n) {
                int v = fact(m);
                int e = exp;
                extGCD.extgcd(fact(n), pc);
                v = modular.mul(v, modular.valueOf(extGCD.getX()));
                e -= exp;
                extGCD.extgcd(fact(m - n), pc);
                v = modular.mul(v, modular.valueOf(extGCD.getX()));
                e -= exp;
                v = modular.mul(v, power.pow(p, e));
                return v;
            }
        }

        /**
         * Extend lucas algorithm
         */
        public static class ExtLucas {
            PollardRho pr = new PollardRho();
            Map<Integer, ExtLucasFactorial> factorialMap = new HashMap();

            public ExtLucas(int p) {
                Map<Integer, Integer> factors = pr.findAllFactors(p);
                for (Map.Entry<Integer, Integer> entry : factors.entrySet()) {
                    factorialMap.put(entry.getValue(), new ExtLucasFactorial(entry.getKey(), entry.getValue(), new int[entry.getValue() + 1]));
                }
            }

            /**
             * Get C(m, n) % p
             */
            public int composite(long m, long n) {
                ExtCRT extCRT = new ExtCRT();
                for (Map.Entry<Integer, ExtLucasFactorial> entry : factorialMap.entrySet()) {
                    extCRT.add(entry.getValue().composite(m, n), entry.getKey());
                }
                return (int) extCRT.r;
            }
        }

        /**
         * Extend lucas algorithm long version
         */
        public static class LongExtLucas {
            LongPollardRho pr = new LongPollardRho();
            Map<Integer, ExtLucasFactorial> factorialMap = new HashMap();

            public LongExtLucas(long p) {
                Map<Long, Long> factors = pr.findAllFactors(p);
                for (Map.Entry<Long, Long> entry : factors.entrySet()) {
                    factorialMap.put(entry.getValue().intValue(), new ExtLucasFactorial(entry.getKey().intValue(), entry.getValue().intValue(), new int[entry.getValue().intValue() + 1]));
                }
            }

            /**
             * Get C(m, n) % p
             */
            public int composite(long m, long n) {
                ExtCRT extCRT = new ExtCRT();
                for (Map.Entry<Integer, ExtLucasFactorial> entry : factorialMap.entrySet()) {
                    extCRT.add(entry.getValue().composite(m, n), entry.getKey());
                }
                return (int) extCRT.r;
            }
        }

        /**
         * Extend gcd
         */
        public static class ExtGCD {
            private long x;
            private long y;
            private long g;

            public long getX() {
                return x;
            }

            public long getY() {
                return y;
            }

            /**
             * Get g = Gcd(a, b) and find a way to set x and y to match ax+by=g
             */
            public long extgcd(long a, long b) {
                if (a >= b) {
                    g = extgcd0(a, b);
                } else {
                    g = extgcd0(b, a);
                    long tmp = x;
                    x = y;
                    y = tmp;
                }
                return g;
            }


            private long extgcd0(long a, long b) {
                if (b == 0) {
                    x = 1;
                    y = 0;
                    return a;
                }
                long g = extgcd0(b, a % b);
                long n = x;
                long m = y;
                x = m;
                y = n - m * (a / b);
                return g;
            }
        }

        public static class Gcd {
            public long gcd(long a, long b) {
                return a >= b ? gcd0(a, b) : gcd0(b, a);
            }

            private long gcd0(long a, long b) {
                return b == 0 ? a : gcd0(b, a % b);
            }

            public int gcd(int a, int b) {
                return a >= b ? gcd0(a, b) : gcd0(b, a);
            }

            private int gcd0(int a, int b) {
                return b == 0 ? a : gcd0(b, a % b);
            }
        }

        /**
         * Euler sieve for filter primes
         */
        public static class EulerSieve {
            int[] primes;
            boolean[] isComp;
            int primeLength;

            public EulerSieve(int limit) {
                isComp = new boolean[limit + 1];
                primes = new int[limit + 1];
                primeLength = 0;
                for (int i = 2; i <= limit; i++) {
                    if (!isComp[i]) {
                        primes[primeLength++] = i;
                    }
                    for (int j = 0, until = limit / i; j < primeLength && primes[j] <= until; j++) {
                        int pi = primes[j] * i;
                        isComp[pi] = true;
                        if (i % primes[j] == 0) {
                            break;
                        }
                    }
                }
            }
        }

        /**
         * Euler sieve for multiplicative function
         */
        public static class MultiplicativeFunctionSieve {
            int[] primes;
            boolean[] isComp;
            int primeLength;
            int[] mobius;
            int[] euler;
            int[] factors;
            int[] smallestPrimeFactor;
            int[] numberOfSmallestPrimeFactor;

            public MultiplicativeFunctionSieve(int limit, boolean enableMobius, boolean enableEuler, boolean enableFactors) {
                isComp = new boolean[limit + 1];
                primes = new int[limit + 1];
                numberOfSmallestPrimeFactor = new int[limit + 1];
                smallestPrimeFactor = new int[limit + 1];
                primeLength = 0;
                for (int i = 2; i <= limit; i++) {
                    if (!isComp[i]) {
                        primes[primeLength++] = i;
                        numberOfSmallestPrimeFactor[i] = smallestPrimeFactor[i] = i;
                    }
                    for (int j = 0, until = limit / i; j < primeLength && primes[j] <= until; j++) {
                        int pi = primes[j] * i;
                        smallestPrimeFactor[pi] = primes[j];
                        numberOfSmallestPrimeFactor[pi] = smallestPrimeFactor[i] == primes[j]
                                ? (numberOfSmallestPrimeFactor[i] * numberOfSmallestPrimeFactor[primes[j]])
                                : numberOfSmallestPrimeFactor[primes[j]];
                        isComp[pi] = true;
                        if (i % primes[j] == 0) {
                            break;
                        }
                    }
                }

                if (enableMobius) {
                    mobius = new int[limit + 1];
                    mobius[1] = 1;
                    for (int i = 2; i <= limit; i++) {
                        if (!isComp[i]) {
                            mobius[i] = -1;
                        } else {
                            if (numberOfSmallestPrimeFactor[i] != smallestPrimeFactor[i]) {
                                mobius[i] = 0;
                            } else {
                                mobius[i] = mobius[numberOfSmallestPrimeFactor[i]] *
                                        mobius[i / numberOfSmallestPrimeFactor[i]];
                            }
                        }
                    }
                }

                if (enableEuler) {
                    euler = new int[limit + 1];
                    euler[1] = 1;
                    for (int i = 2; i <= limit; i++) {
                        if (!isComp[i]) {
                            euler[i] = i - 1;
                        } else {
                            if (numberOfSmallestPrimeFactor[i] == i) {
                                euler[i] = i - i / smallestPrimeFactor[i];
                            } else {
                                euler[i] = euler[numberOfSmallestPrimeFactor[i]] *
                                        euler[i / numberOfSmallestPrimeFactor[i]];
                            }
                        }
                    }
                }

                if (enableFactors) {
                    factors = new int[limit + 1];
                    factors[1] = 1;
                    for (int i = 2; i <= limit; i++) {
                        if (!isComp[i]) {
                            factors[i] = 2;
                        } else {
                            if (numberOfSmallestPrimeFactor[i] == i) {
                                factors[i] = 1 + factors[i / smallestPrimeFactor[i]];
                            } else {
                                factors[i] = factors[numberOfSmallestPrimeFactor[i]] *
                                        factors[i / numberOfSmallestPrimeFactor[i]];
                            }
                        }
                    }
                }
            }
        }

        /**
         * Mod operations
         */
        public static class Modular {
            int m;

            public Modular(int m) {
                this.m = m;
            }

            public int valueOf(int x) {
                x %= m;
                if (x < 0) {
                    x += m;
                }
                return x;
            }

            public int valueOf(long x) {
                x %= m;
                if (x < 0) {
                    x += m;
                }
                return (int) x;
            }

            public int mul(int x, int y) {
                return valueOf((long) x * y);
            }

            public int mul(long x, long y) {
                x = valueOf(x);
                y = valueOf(y);
                return valueOf(x * y);
            }

            public int plus(int x, int y) {
                return valueOf(x + y);
            }

            public int plus(long x, long y) {
                x = valueOf(x);
                y = valueOf(y);
                return valueOf(x + y);
            }

            public int subtract(int x, int y) {
                return valueOf(x - y);
            }

            public int subtract(long x, long y) {
                return valueOf(x - y);
            }

            @Override
            public String toString() {
                return "mod " + m;
            }
        }

        /**
         * Bit operations
         */
        public static class BitOperator {
            public int bitAt(int x, int i) {
                return (x >> i) & 1;
            }

            public int bitAt(long x, int i) {
                return (int) ((x >> i) & 1);
            }

            public int setBit(int x, int i, boolean v) {
                if (v) {
                    x |= 1 << i;
                } else {
                    x &= ~(1 << i);
                }
                return x;
            }

            public long setBit(long x, int i, boolean v) {
                if (v) {
                    x |= 1L << i;
                } else {
                    x &= ~(1L << i);
                }
                return x;
            }

            /**
             * Determine whether x is subset of y
             */
            public boolean subset(long x, long y) {
                return intersect(x, y) == x;
            }

            /**
             * Merge two set
             */
            public long merge(long x, long y) {
                return x | y;
            }

            public long intersect(long x, long y) {
                return x & y;
            }

            public long differ(long x, long y) {
                return x - intersect(x, y);
            }
        }

        /**
         * Power operations
         */
        public static class Power {
            public Modular getModular() {
                return modular;
            }

            final Modular modular;

            public Power(Modular modular) {
                this.modular = modular;
            }

            public int pow(int x, long n) {
                if (n == 0) {
                    return 1;
                }
                long r = pow(x, n >> 1);
                r = modular.valueOf(r * r);
                if ((n & 1) == 1) {
                    r = modular.valueOf(r * x);
                }
                return (int) r;
            }

            public int inverse(int x) {
                return pow(x, modular.m - 2);
            }

            public int pow2(int x) {
                return x * x;
            }

            public long pow2(long x) {
                return x * x;
            }

            public double pow2(double x) {
                return x * x;
            }
        }

        /**
         * Log operations
         */
        public static class Log2 {
            public int ceilLog(int x) {
                return 32 - Integer.numberOfLeadingZeros(x - 1);
            }

            public int floorLog(int x) {
                return 31 - Integer.numberOfLeadingZeros(x);
            }

            public int ceilLog(long x) {
                return 64 - Long.numberOfLeadingZeros(x - 1);
            }

            public int floorLog(long x) {
                return 63 - Long.numberOfLeadingZeros(x);
            }
        }

        /**
         * Find all inverse number
         */
        public static class InverseNumber {
            int[] inv;

            public InverseNumber(int[] inv, int limit, Modular modular) {
                this.inv = inv;
                inv[1] = 1;
                int p = modular.m;
                for (int i = 2; i <= limit; i++) {
                    int k = p / i;
                    int r = p % i;
                    inv[i] = modular.mul(-k, inv[r]);
                }
            }

            public InverseNumber(int limit, Modular modular) {
                this(new int[limit + 1], limit, modular);
            }
        }

        /**
         * Factorial
         */
        public static class Factorial {
            int[] fact;
            int[] inv;

            public Factorial(int[] fact, int[] inv, InverseNumber in, int limit, Modular modular) {
                this.fact = fact;
                this.inv = inv;
                fact[0] = inv[0] = 1;
                for (int i = 1; i <= limit; i++) {
                    fact[i] = modular.mul(fact[i - 1], i);
                    inv[i] = modular.mul(inv[i - 1], in.inv[i]);
                }
            }

            public Factorial(int limit, Modular modular) {
                this(new int[limit + 1], new int[limit + 1], new InverseNumber(limit, modular), limit, modular);
            }
        }

        /**
         * Composition
         */
        public static class Composite {
            final Factorial factorial;
            final Modular modular;

            public Composite(Factorial factorial, Modular modular) {
                this.factorial = factorial;
                this.modular = modular;
            }

            public Composite(int limit, Modular modular) {
                this(new Factorial(limit, modular), modular);
            }

            public int composite(int m, int n) {
                if (n > m) {
                    return 0;
                }
                return modular.mul(modular.mul(factorial.fact[m], factorial.inv[n]), factorial.inv[m - n]);
            }
        }

        /**
         * Test whether a number is primes
         */
        public static class MillerRabin {
            Modular modular;
            Power power;

            /**
             * Check whether n is a prime s times
             */
            public boolean mr(int n, int s) {
                if (n == 2) {
                    return true;
                }
                if (n % 2 == 0) {
                    return false;
                }
                modular = new Modular(n);
                power = new Power(modular);
                for (int i = 0; i < s; i++) {
                    int x = RANDOM.nextInt(n - 2) + 2;
                    if (!mr0(x, n)) {
                        return false;
                    }
                }
                return true;
            }

            private boolean mr0(int x, int n) {
                int exp = n - 1;
                while (true) {
                    int y = power.pow(x, exp);
                    if (y != 1 && y != n - 1) {
                        return false;
                    }
                    if (y != 1 || exp % 2 == 1) {
                        break;
                    }
                    exp = exp / 2;
                }
                return true;
            }
        }

        /**
         * Modular operation for long version
         */
        public static class LongModular {
            final long m;

            public LongModular(long m) {
                this.m = m;
            }

            public long mul(long a, long b) {
                return b == 0 ? 0 : ((mul(a, b >> 1) << 1) % m + a * (b & 1)) % m;
            }

            public long plus(long a, long b) {
                return valueOf(a + b);
            }

            public long valueOf(long a) {
                a %= m;
                if (a < 0) {
                    a += m;
                }
                return a;
            }
        }

        public static class LongPower {
            final LongModular modular;

            public LongPower(LongModular modular) {
                this.modular = modular;
            }

            long pow(long x, long n) {
                if (n == 0) {
                    return 1;
                }
                long r = pow(x, n >> 1);
                r = modular.mul(r, r);
                if ((n & 1) == 1) {
                    r = modular.mul(r, x);
                }
                return r;
            }

            long inverse(long x) {
                return pow(x, modular.m - 2);
            }
        }

        /**
         * Test whether a number is primes
         */
        public static class LongMillerRabin {
            LongModular modular;
            LongPower power;

            /**
             * Check whether n is a prime s times
             */
            public boolean mr(long n, int s) {
                if (n == 2) {
                    return true;
                }
                if (n % 2 == 0) {
                    return false;
                }
                modular = new LongModular(n);
                power = new LongPower(modular);
                for (int i = 0; i < s; i++) {
                    long x = (long) (RANDOM.nextDouble() * (n - 2) + 2);
                    if (!mr0(x, n)) {
                        return false;
                    }
                }
                return true;
            }

            private boolean mr0(long x, long n) {
                long exp = n - 1;
                while (true) {
                    long y = power.pow(x, exp);
                    if (y != 1 && y != n - 1) {
                        return false;
                    }
                    if (y != 1 || exp % 2 == 1) {
                        break;
                    }
                    exp = exp / 2;
                }
                return true;
            }
        }

        public static class LongPollardRho {
            LongMillerRabin mr = new LongMillerRabin();
            Gcd gcd = new Gcd();
            LongModular modular;

            /**
             * Find a factor of n, if n is returned, it means n is 1 or a prime
             */
            public long findFactor(long n) {
                if (mr.mr(n, 3)) {
                    return n;
                }
                modular = new LongModular(n);
                while (true) {
                    long f = findFactor0((long) (RANDOM.nextDouble() * n), (long) (RANDOM.nextDouble() * n), n);
                    if (f != -1) {
                        return f;
                    }
                }
            }

            private long findFactor0(long x, long c, long n) {
                long xi = x;
                long xj = x;
                int j = 2;
                int i = 1;
                while (i < n) {
                    i++;
                    xi = modular.plus(modular.mul(xi, xi), c);
                    long g = gcd.gcd(n, Math.abs(xi - xj));
                    if (g != 1 && g != n) {
                        return g;
                    }
                    if (i == j) {
                        j = j << 1;
                        xj = xi;
                    }
                }
                return -1;
            }

            /**
             * Find the representation of n=p1^c1 * p2^c2 * ... * pm ^ cm.
             * <br>
             * The returned map contained such entries: pi -> pi^ci
             */
            public Map<Long, Long> findAllFactors(long n) {
                Map<Long, Long> map = new HashMap();
                findAllFactors(map, n);
                return map;
            }

            private void findAllFactors(Map<Long, Long> map, long n) {
                if (n == 1) {
                    return;
                }
                long f = findFactor(n);
                if (f == n) {
                    Long value = map.get(f);
                    if (value == null) {
                        value = 1L;
                    }
                    map.put(f, value * f);
                    return;
                }
                findAllFactors(map, f);
                findAllFactors(map, n / f);
            }

        }

        /**
         * Extend chinese remainder theory
         */
        public static class ExtCRT {
            /**
             * remainder
             */
            long r;
            /**
             * modulus
             */
            long m;
            ExtGCD gcd = new ExtGCD();

            public ExtCRT() {
                r = 0;
                m = 1;
            }

            /**
             * Add a new condition: x % m = r
             */
            public boolean add(long r, long m) {
                long m1 = this.m;
                long x1 = this.r;
                long m2 = m;
                long x2 = ((r % m) + m) % m;
                long g = gcd.extgcd(m1, m2);
                long a = gcd.getX();
                if ((x2 - x1) % g != 0) {
                    return false;
                }
                this.m = m1 / g * m2;
                this.r = BigInteger.valueOf(a).multiply(BigInteger.valueOf((x2 - x1) / g))
                        .multiply(BigInteger.valueOf(m1)).add(BigInteger.valueOf(x1))
                        .mod(BigInteger.valueOf(this.m)).longValue();
                return true;
            }
        }

        /**
         * Lucas algorithm
         */
        public static class Lucas {
            private final Composite composite;
            private int modulus;

            public Lucas(Composite composite) {
                this.composite = composite;
                this.modulus = composite.modular.m;
            }

            public int composite(long m, long n) {
                if (n == 0) {
                    return 1;
                }
                return composite.modular.mul(composite.composite((int) (m % modulus), (int) (n % modulus)),
                        composite(m / modulus, n / modulus));
            }
        }

        /**
         * Find all factors of a number
         */
        public static class PollardRho {
            MillerRabin mr = new MillerRabin();
            Gcd gcd = new Gcd();
            Random random = new Random();

            public int findFactor(int n) {
                if (mr.mr(n, 10)) {
                    return n;
                }
                while (true) {
                    int f = findFactor0(random.nextInt(n), random.nextInt(n), n);
                    if (f != -1) {
                        return f;
                    }
                }
            }

            /**
             * Find all prime factor of n
             * <br>
             * p1 => p1^c1
             * <br>
             * ...
             * <br>
             * pk => pk^ck
             */
            public Map<Integer, Integer> findAllFactors(int n) {
                Map<Integer, Integer> map = new HashMap();
                findAllFactors(map, n);
                return map;
            }

            private void findAllFactors(Map<Integer, Integer> map, int n) {
                if (n == 1) {
                    return;
                }
                int f = findFactor(n);
                if (f == n) {
                    Integer value = map.get(f);
                    if (value == null) {
                        value = 1;
                    }
                    map.put(f, value * f);
                    return;
                }
                findAllFactors(map, f);
                findAllFactors(map, n / f);
            }

            private int findFactor0(int x, int c, int n) {
                int xi = x;
                int xj = x;
                int j = 2;
                int i = 1;
                while (i < n) {
                    i++;
                    xi = (int) ((long) xi * xi + c) % n;
                    int g = gcd.gcd(n, Math.abs(xi - xj));
                    if (g != 1 && g != n) {
                        return g;
                    }
                    if (i == j) {
                        j = j << 1;
                        xj = xi;
                    }
                }
                return -1;
            }
        }

        public static class ModExpression {
            ExtGCD extGCD = new ExtGCD();
            Modular modular = new Modular(1);

            /**
             * Find ka=b(mod c) where k is the minimum possible non negative integer.
             * <br>
             * If it's impossible, -1 will be returned.
             */
            public long solve(long a, long b, int c) {
                modular.m = c;
                a = modular.valueOf(a);
                b = modular.valueOf(b);
                int g = (int) extGCD.extgcd((int) a, c);
                if (b % g != 0) {
                    return -1;
                }
                modular.m = c / g;
                return modular.valueOf(b / g * extGCD.getX());
            }
        }

        /**
         * \sum_{i=1}^{limit}f(\lfloor n/i \rfloor)
         */
        public static class FloorDivisionOptimizer {
            int l;
            int r;
            int n;
            int limit;


            public FloorDivisionOptimizer(int n, int l, int limit) {
                this.n = n;
                this.l = 0;
                this.limit = limit;
                this.r = l - 1;
            }

            public boolean hasNext() {
                return r < limit;
            }

            public int next() {
                l = r + 1;
                r = n / (n / l);
                return n / l;
            }
        }


        public static class QuadraticResidue {
            final Modular modular;
            final BitOperator bitOperator = new BitOperator();
            Power power;
            final PollardRho rho = new PollardRho();


            public QuadraticResidue(Modular modular) {
                this.modular = modular;
                power = new Power(modular);
            }

            /**
             * return \sqrt{n} or -1 if it doesn't exist
             */
            public int square(int n) {
                n = modular.valueOf(n);
                if (n == 0) {
                    return 0;
                }
                int p = modular.m;
                if (power.pow(n, (p - 1) / 2) != 1) {
                    return -1;
                }
                while (true) {
                    int a = RANDOM.nextInt(p);
                    int w = modular.plus(modular.mul(a, a), -n);
                    if (power.pow(w, (p - 1) / 2) == 1) {
                        continue;
                    }


                    int pow = (p + 1) / 2;
                    int i = 31 - Integer.numberOfLeadingZeros(pow);
                    int real = 1;
                    int img = 0;
                    for (; i >= 0; i--) {
                        int nReal = modular.plus(modular.mul(real, real),
                                modular.mul(modular.mul(img, img), w));
                        int nImg = modular.mul(modular.mul(real, img), 2);
                        real = nReal;
                        img = nImg;
                        if (bitOperator.bitAt(pow, i) == 1) {
                            nReal = modular.plus(modular.mul(real, a), modular.mul(img, w));
                            nImg = modular.plus(modular.mul(img, a), real);
                            real = nReal;
                            img = nImg;
                        }
                    }

                    return real;
                }
            }

            public int minPrimitiveRoot() {
                if (modular.m == 2) {
                    return 1;
                }
                Map<Integer, Integer> factorMap = rho.findAllFactors(modular.m - 1);
                int[] factors = factorMap.keySet().stream().mapToInt(Integer::intValue).toArray();
                for (int i = 2; ; i++) {
                    boolean valid = true;
                    for (int factor : factors) {
                        if (power.pow(i, (modular.m - 1) / factor) == 1) {
                            valid = false;
                            break;
                        }
                    }
                    if (valid) {
                        return i;
                    }
                }
            }
        }
    }

    public static class NumberTheoryTransform {
        private static final NumberTheory.Modular MODULAR = new NumberTheory.Modular(998244353);
        private static final NumberTheory.Power POWER = new NumberTheory.Power(MODULAR);
        private static final int G = 3;
        private static int[] wCache = new int[23];
        private static int[] invCache = new int[23];

        static {
            for (int i = 0, until = wCache.length; i < until; i++) {
                int s = 1 << i;
                wCache[i] = POWER.pow(G, (MODULAR.m - 1) / 2 / s);
                invCache[i] = POWER.inverse(s);
            }
        }

        public static void dotMul(int[] a, int[] b, int[] c, int m) {
            for (int i = 0, n = 1 << m; i < n; i++) {
                c[i] = MODULAR.mul(a[i], b[i]);
            }
        }

        public static void prepareReverse(int[] r, int b) {
            int n = 1 << b;
            r[0] = 0;
            for (int i = 1; i < n; i++) {
                r[i] = (r[i >> 1] >> 1) | ((1 & i) << (b - 1));
            }
        }

        private static void dft(int[] r, int[] p, int m) {
            int n = 1 << m;

            for (int i = 0; i < n; i++) {
                if (r[i] > i) {
                    int tmp = p[i];
                    p[i] = p[r[i]];
                    p[r[i]] = tmp;
                }
            }

            int w = 0;
            int t = 0;
            for (int d = 0; d < m; d++) {
                int w1 = wCache[d];
                int s = 1 << d;
                int s2 = s << 1;
                for (int i = 0; i < n; i += s2) {
                    w = 1;
                    for (int j = 0; j < s; j++) {
                        int a = i + j;
                        int b = a + s;
                        t = MODULAR.mul(w, p[b]);
                        p[b] = MODULAR.plus(p[a], -t);
                        p[a] = MODULAR.plus(p[a], t);
                        w = MODULAR.mul(w, w1);
                    }
                }
            }
        }

        private static void idft(int[] r, int[] p, int m) {
            dft(r, p, m);

            int n = 1 << m;
            int invN = invCache[m];

            p[0] = MODULAR.mul(p[0], invN);
            p[n / 2] = MODULAR.mul(p[n / 2], invN);
            for (int i = 1, until = n / 2; i < until; i++) {
                int a = p[n - i];
                p[n - i] = MODULAR.mul(p[i], invN);
                p[i] = MODULAR.mul(a, invN);
            }
        }

        public static void reverse(int[] p, int l, int r) {
            while (l < r) {
                int tmp = p[l];
                p[l] = p[r];
                p[r] = tmp;
                l++;
                r--;
            }
        }

        public static int rankOf(int[] p) {
            for (int i = p.length - 1; i >= 0; i--) {
                if (p[i] > 0) {
                    return i;
                }
            }
            return 0;
        }

        /**
         * calc a = b * c + remainder, m >= 1 + ceil(log_2 max(|a|, |b|))
         */
        public static void divide(int[] r, int[] a, int[] b, int[] c, int[] remainder, int m) {
            int rankA = rankOf(a);
            int rankB = rankOf(b);
            reverse(a, 0, rankA);
            reverse(b, 0, rankB);
            inverse(r, b, c, remainder, m - 1);
            dft(r, a, m);
            dft(r, c, m);
            dotMul(a, c, c, m);
            idft(r, a, m);
            idft(r, c, m);
            reverse(a, 0, rankA);
            reverse(b, 0, rankB);
            for (int i = rankA - rankB + 1; i < c.length; i++) {
                c[i] = 0;
            }
            reverse(c, 0, rankA - rankB);

            dft(r, a, m);
            dft(r, b, m);
            dft(r, c, m);
            for (int i = 0; i < remainder.length; i++) {
                remainder[i] = MODULAR.subtract(a[i], MODULAR.mul(b[i], c[i]));
            }
            idft(r, a, m);
            idft(r, b, m);
            idft(r, c, m);
            idft(r, remainder, m);
        }

        /**
         * return polynomial g while p * g = 1 (mod x^m).
         * <br>
         * You are supposed to guarantee the lengths of all arrays are greater than or equal to 2^{ceil(log2(m)) + 1}
         */
        private static void inverse(int[] r, int[] p, int[] inv, int[] buf, int m) {
            if (m == 0) {
                inv[0] = POWER.inverse(p[0]);
                return;
            }
            inverse(r, p, inv, buf, m - 1);
            int n = 1 << (m + 1);
            System.arraycopy(p, 0, buf, 0, 1 << m);
            Arrays.fill(buf, 1 << m, 1 << (m + 1), 0);
            prepareReverse(r, (m + 1));
            dft(r, buf, (m + 1));
            dft(r, inv, (m + 1));
            for (int i = 0; i < n; i++) {
                inv[i] = MODULAR.mul(inv[i], 2 - MODULAR.mul(buf[i], inv[i]));
            }
            idft(r, inv, m + 1);
            for (int i = 1 << m; i < n; i++) {
                inv[i] = 0;
            }
        }
    }
}