Problem: Julia is going to cook a chicken in the kitchen of her dormitory. To save energy, the stove in the kitchen automatically turns off after k minutes after turning on.

During cooking, Julia goes to the kitchen every d minutes and turns on the stove if it is turned off. While the cooker is turned off, it stays warm. The stove switches on and off instantly.

It is known that the chicken needs t minutes to be cooked on the stove, if it is turned on, and 2t minutes, if it is turned off. You need to find out, how much time will Julia have to cook the chicken, if it is considered that the chicken is cooked evenly, with constant speed when the stove is turned on and at a constant speed when it is turned off.

Input

The single line contains three integers k, d and t (1 ≤ k, d, t ≤ 1018).

Output

Print a single number, the total time of cooking in minutes. The relative or absolute error must not exceed 10 - 9.

Namely, let's assume that your answer is x and the answer of the jury is y. The checker program will consider your answer correct if <image>.

Examples

Input

3 2 6


Output

6.5


Input

4 2 20


Output

20.0

Note

In the first example, the chicken will be cooked for 3 minutes on the turned on stove, after this it will be cooked for <image>. Then the chicken will be cooked for one minute on a turned off stove, it will be cooked for <image>. Thus, after four minutes the chicken will be cooked for <image>. Before the fifth minute Julia will turn on the stove and after 2.5 minutes the chicken will be ready <image>.

In the second example, when the stove is turned off, Julia will immediately turn it on, so the stove will always be turned on and the chicken will be cooked in 20 minutes.

Solution (in JAVA):

Solution:
//package baobab;

import java.io.*;
import java.util.*;

public class C {

    public static final boolean DEBUG_PRINTS = false;

    public static void main(String[] args) {
        Solver solver = new Solver();
    }

    static class Solver {
        IO io;

        public Solver() {
            this.io = new IO();
            try {
                solve();
            } catch (RuntimeException e) {
                if (!e.getMessage().equals("Clean exit")) {
                    throw e;
                }
            } finally {
                io.close();
            }
        }

        /****************************** START READING HERE ********************************/

        void solve() {
            long k = io.nextLong();
            long d = io.nextLong();
            long halfMinutesSpent = 0;
            long cookUnitsOverall = 2*io.nextLong();
            long cookUnitsLeft = cookUnitsOverall;

            boolean stoveIsOn = true;
            long timeWhenStoveOffNext = k;
            long timeWhenJuliaNext = (k/d)*d + (k%d > 0 ? d : 0);
            long startStateStove = timeWhenStoveOffNext;
            long startStateJulia = timeWhenJuliaNext;
            // loop detected when startstate encountered again

            PriorityQueue<Event> events = new PriorityQueue<>();
            int ID_JULIA_RETURNS = 1;
            int ID_STOVE_GOES_OFF = 2;
            events.add(new Event(ID_JULIA_RETURNS, timeWhenJuliaNext));
            events.add(new Event(ID_STOVE_GOES_OFF, timeWhenStoveOffNext));

            long prevTime = 0;
            while (!events.isEmpty()) {
                Event e = events.poll();
                long halfMinutesThisInterval = 2*(e.time - prevTime);
                prevTime = e.time;
                halfMinutesSpent += halfMinutesThisInterval;
                long cookUnitsThisInterval = (stoveIsOn ? halfMinutesThisInterval : halfMinutesThisInterval/2);
                //io.println("Event " + e.id + " at time " + e.time + " cookUnitsLeft " + cookUnitsLeft + " then cookUnitsThisInterval " + cookUnitsThisInterval);
                if (cookUnitsThisInterval >= cookUnitsLeft) {
                    // avoid overcooking & break if we are finished
                    halfMinutesSpent -= halfMinutesThisInterval;
                    halfMinutesSpent += (stoveIsOn ? cookUnitsLeft : cookUnitsLeft*2);
                    break;
                }
                cookUnitsLeft -= cookUnitsThisInterval;

                if (e.id == ID_JULIA_RETURNS) {
                    if (!stoveIsOn) {
                        stoveIsOn = true;
                        timeWhenStoveOffNext = e.time + k;
                        events.add(new Event(ID_STOVE_GOES_OFF, timeWhenStoveOffNext));

                        long minutesToStoveOffNext = timeWhenStoveOffNext - e.time;
                        timeWhenJuliaNext = e.time + ((minutesToStoveOffNext/d)*d + (minutesToStoveOffNext%d > 0 ? d : 0));
                        events.add(new Event(ID_JULIA_RETURNS, timeWhenJuliaNext));
                    }
                } else if (e.id == ID_STOVE_GOES_OFF) {
                    stoveIsOn = false;
                }

                // if we are in same state as beginning, we have completed a loop
                long minutesToStoveOffNext = timeWhenStoveOffNext - e.time;
                long minutesToJuliaNext = timeWhenJuliaNext - e.time;
                //io.println("    state: " + stoveIsOn + " minutesToStoveOff " + minutesToStoveOffNext + " minutesToJuliaNext " + minutesToJuliaNext);
                if (stoveIsOn && minutesToStoveOffNext == startStateStove && minutesToJuliaNext == startStateJulia) {
                    // loop detected. perform n more full loops (where n can be 0)
                    long cookUnitsInLoop = cookUnitsOverall - cookUnitsLeft;
                    long halfMinutesInLoop = halfMinutesSpent;
                    long fullLoopsToStillDo = cookUnitsLeft / cookUnitsInLoop;
                    halfMinutesSpent += (fullLoopsToStillDo * halfMinutesInLoop);
                    cookUnitsLeft -= (fullLoopsToStillDo * cookUnitsInLoop);
                }
            }



            String ans = "" + halfMinutesSpent/2;
            if (halfMinutesSpent%2!=0) {
                ans += ".5";
            }
            io.println(ans);
        }

        /************************** UTILITY CODE BELOW THIS LINE **************************/

        long MOD = (long)1e9 + 7;

        List<Integer>[] toGraph(IO io, int n) {
            List<Integer>[] g = new ArrayList[n+1];
            for (int i=1; i<=n; i++) g[i] = new ArrayList<>();
            for (int i=1; i<=n-1; i++) {
                int a = io.nextInt();
                int b = io.nextInt();
                g[a].add(b);
                g[b].add(a);
            }
            return g;
        }

        class Point {
            int y;
            int x;

            public Point(int y, int x) {
                this.y = y;
                this.x = x;
            }
        }

        class Event implements Comparable<Event> {
            int id;
            long time;

            public Event(int id, long time) {
                this.time = time;
                this.id = id;
            }

            @Override
            public int compareTo(Event o) {
                if (this.time < o.time) return -1;
                if (this.time > o.time) return 1;
                if (this.id == 2) return -1;
                else return 1;
            }
        }

        long pow(long base, int exp) {
            if (exp == 0) return 1L;
            long x = pow(base, exp/2);
            long ans = x * x;
            if (exp % 2 != 0) ans *= base;
            return ans;
        }

        long gcd(long... v) {
            /** Chained calls to Euclidean algorithm. */
            if (v.length == 1) return v[0];
            long ans = gcd(v[1], v[0]);
            for (int i=2; i<v.length; i++) {
                ans = gcd(ans, v[i]);
            }
            return ans;
        }

        long gcd(long a, long b) {
            /** Euclidean algorithm. */
            if (b == 0) return a;
            return gcd(b, a%b);
        }

        int[] generatePrimesUpTo(int last) {
            /* Sieve of Eratosthenes. Practically O(n). Values of 0 indicate primes. */
            int[] div = new int[last+1];
            for (int x=2; x<=last; x++) {
                if (div[x] > 0) continue;
                for (int u=2*x; u<=last; u+=x) {
                    div[u] = x;
                }
            }
            return div;
        }

        long lcm(long a, long b) {
            /** Least common multiple */
            return a * b / gcd(a,b);
        }

        private class ElementCounter {
            private HashMap<Long, Integer> elements;

            public ElementCounter() {
                elements = new HashMap<>();
            }

            public void add(long element) {
                int count = 1;
                if (elements.containsKey(element)) count += elements.get(element);
                elements.put(element, count);
            }

            public void remove(long element) {
                int count = elements.get(element);
                count--;
                if (count == 0) elements.remove(element);
                else elements.put(element, count);
            }

            public int get(long element) {
                if (!elements.containsKey(element)) return 0;
                return elements.get(element);
            }

            public int size() {
                return elements.size();
            }
        }

        class StringCounter {
            HashMap<String, Long> elements;

            public StringCounter() {
                elements = new HashMap<>();
            }

            public void add(String identifier) {
                long count = 1;
                if (elements.containsKey(identifier)) count += elements.get(identifier);
                elements.put(identifier, count);
            }

            public void remove(String identifier) {
                long count = elements.get(identifier);
                count--;
                if (count == 0) elements.remove(identifier);
                else elements.put(identifier, count);
            }

            public long get(String identifier) {
                if (!elements.containsKey(identifier)) return 0;
                return elements.get(identifier);
            }

            public int size() {
                return elements.size();
            }
        }

        class DisjointSet {
            /** Union Find / Disjoint Set data structure. */
            int[] size;
            int[] parent;
            int componentCount;

            public DisjointSet(int n) {
                componentCount = n;
                size = new int[n];
                parent = new int[n];
                for (int i=0; i<n; i++) parent[i] = i;
                for (int i=0; i<n; i++) size[i] = 1;
            }

            public void join(int a, int b) {
                /* Find roots */
                int rootA = parent[a];
                int rootB = parent[b];
                while (rootA != parent[rootA]) rootA = parent[rootA];
                while (rootB != parent[rootB]) rootB = parent[rootB];

                if (rootA == rootB) {
                    /* Already in the same set */
                    return;
                }

                /* Merge smaller set into larger set. */
                if (size[rootA] > size[rootB]) {
                    size[rootA] += size[rootB];
                    parent[rootB] = rootA;
                } else {
                    size[rootB] += size[rootA];
                    parent[rootA] = rootB;
                }
                componentCount--;
            }

        }

        class LCAFinder {

            /* O(n log n) Initialize: new LCAFinder(graph)
             * O(log n) Queries: find(a,b) returns lowest common ancestor for nodes a and b */

            int[] nodes;
            int[] depths;
            int[] entries;
            int pointer;
            FenwickMin fenwick;

            public LCAFinder(List<Integer>[] graph) {
                this.nodes = new int[(int)10e6];
                this.depths = new int[(int)10e6];
                this.entries = new int[graph.length];
                this.pointer = 1;
                boolean[] visited = new boolean[graph.length+1];
                dfs(1, 0, graph, visited);
                fenwick = new FenwickMin(pointer-1);
                for (int i=1; i<pointer; i++) {
                    fenwick.set(i, depths[i] * 1000000L + i);
                }
            }

            private void dfs(int node, int depth, List<Integer>[] graph, boolean[] visited) {
                visited[node] = true;
                entries[node] = pointer;
                nodes[pointer] = node;
                depths[pointer] = depth;
                pointer++;
                for (int neighbor : graph[node]) {
                    if (visited[neighbor]) continue;
                    dfs(neighbor, depth+1, graph, visited);
                    nodes[pointer] = node;
                    depths[pointer] = depth;
                    pointer++;
                }
            }

            public int find(int a, int b) {
                int left = entries[a];
                int right = entries[b];
                if (left > right) {
                    int temp = left;
                    left = right;
                    right = temp;
                }
                long mixedBag = fenwick.getMin(left, right);
                int index = (int) (mixedBag % 1000000L);
                return nodes[index];
            }
        }

        class FenwickMin {
            long n;
            long[] original;
            long[] bottomUp;
            long[] topDown;

            public FenwickMin(int n) {
                this.n = n;
                original = new long[n+2];
                bottomUp = new long[n+2];
                topDown = new long[n+2];
            }

            public void set(int modifiedNode, long value) {
                long replaced = original[modifiedNode];
                original[modifiedNode] = value;
                // Update left tree
                int i = modifiedNode;
                long v = value;
                while (i <= n) {
                    if (v > bottomUp[i]) {
                        if (replaced == bottomUp[i]) {
                            v = Math.min(v, original[i]);
                            for (int r=1 ;; r++) {
                                int x = (i&-i)>>>r;
                                if (x == 0) break;
                                int child = i-x;
                                v = Math.min(v, bottomUp[child]);
                            }
                        } else break;
                    }
                    if (v == bottomUp[i]) break;
                    bottomUp[i] = v;
                    i += (i&-i);
                }
                // Update right tree
                i = modifiedNode;
                v = value;
                while (i > 0) {
                    if (v > topDown[i]) {
                        if (replaced == topDown[i]) {
                            v = Math.min(v, original[i]);
                            for (int r=1 ;; r++) {
                                int x = (i&-i)>>>r;
                                if (x == 0) break;
                                int child = i+x;
                                if (child > n+1) break;
                                v = Math.min(v, topDown[child]);
                            }
                        } else break;
                    }
                    if (v == topDown[i]) break;
                    topDown[i] = v;
                    i -= (i&-i);
                }
            }

            public long getMin(int a, int b) {
                long min = original[a];
                int prev = a;
                int curr = prev + (prev&-prev); // parent right hand side
                while (curr <= b) {
                    min = Math.min(min, topDown[prev]); // value from the other tree
                    prev = curr;
                    curr = prev + (prev&-prev);;
                }
                min = Math.min(min, original[prev]);
                prev = b;
                curr = prev - (prev&-prev); // parent left hand side
                while (curr >= a) {
                    min = Math.min(min,bottomUp[prev]); // value from the other tree
                    prev = curr;
                    curr = prev - (prev&-prev);
                }
                return min;
            }

        }

        class FenwickSum {
            public long[] d;

            public FenwickSum(int n) {
                d=new long[n+1];
            }

            /** a[0] must be unused. */
            public FenwickSum(long[] a) {
                d=new long[a.length];
                for (int i=1; i<a.length; i++) {
                    modify(i, a[i]);
                }
            }

            /** Do not modify i=0. */
            void modify(int i, long v) {
                while (i<d.length) {
                    d[i] += v;
                    // Move to next uplink on the RIGHT side of i
                    i += (i&-i);
                }
            }

            /** Returns sum from a to b, *BOTH* inclusive. */
            long getSum(int a, int b) {
                return getSum(b) - getSum(a-1);
            }

            private long getSum(int i) {
                long sum = 0;
                while (i>0) {
                    sum += d[i];
                    // Move to next uplink on the LEFT side of i
                    i -= (i&-i);
                }
                return sum;
            }
        }

        class SegmentTree {
            /** Query sums with log(n) modifyRange */
            int N;
            long[] p;

            public SegmentTree(int n) {
                /* TODO: Test that this works. */
                for (N=2; N<n; N++) N *= 2;
                p = new long[2*N];
            }

            public void modifyRange(int a, int b, long change) {
                muuta(a, change);
                muuta(b+1, -change);
            }

            void muuta(int k, long muutos) {
                k += N;
                p[k] += muutos;
                for (k /= 2; k >= 1; k /= 2) {
                    p[k] = p[2*k] + p[2*k+1];
                }
            }

            public long get(int k) {
                int a = N;
                int b = k+N;
                long s = 0;
                while (a <= b) {
                    if (a%2 == 1) s += p[a++];
                    if (b%2 == 0) s += p[b--];
                    a /= 2;
                    b /= 2;
                }
                return s;
            }

        }

        class Zalgo {

            public int pisinEsiintyma(String haku, String kohde) {
                char[] s = new char[haku.length() + 1 + kohde.length()];
                for (int i=0; i<haku.length(); i++) {
                    s[i] = haku.charAt(i);
                }
                int j = haku.length();
                s[j++] = '#';
                for (int i=0; i<kohde.length(); i++) {
                    s[j++] = kohde.charAt(i);
                }
                int[] z = toZarray(s);
                int max = 0;
                for (int i=haku.length(); i<z.length; i++) {
                    max = Math.max(max, z[i]);
                }
                return max;
            }

            public int[] toZarray(char[] s) {
                int n = s.length;
                int[] z = new int[n];
                int a = 0, b = 0;
                for (int i = 1; i < n; i++) {
                    if (i > b) {
                        for (int j = i; j < n && s[j - i] == s[j]; j++)  z[i]++;
                    }
                    else {
                        z[i] = z[i - a];
                        if (i + z[i - a] > b) {
                            for (int j = b + 1; j < n && s[j - i] == s[j]; j++) z[i]++;
                            a = i;
                            b = i + z[i] - 1;
                        }
                    }
                }
                return z;
            }

            public List<Integer> getStartIndexesWhereWordIsFound(String haku, String kohde) {
                // this is alternative use case
                char[] s = new char[haku.length() + 1 + kohde.length()];
                for (int i=0; i<haku.length(); i++) {
                    s[i] = haku.charAt(i);
                }
                int j = haku.length();
                s[j++] = '#';
                for (int i=0; i<kohde.length(); i++) {
                    s[j++] = kohde.charAt(i);
                }
                int[] z = toZarray(s);
                List<Integer> indexes = new ArrayList<>();
                for (int i=haku.length(); i<z.length; i++) {
                    if (z[i] < haku.length()) continue;
                    indexes.add(i);
                }
                return indexes;
            }

        }

        class StringHasher {

            class HashedString {
                long[] hashes;
                long[] modifiers;

                public HashedString(long[] hashes, long[] modifiers) {
                    this.hashes = hashes;
                    this.modifiers = modifiers;
                }
            }

            long P;
            long M;

            public StringHasher() {
                initializePandM();
            }

            HashedString hashString(String s) {
                int n = s.length();
                long[] hashes = new long[n];
                long[] modifiers = new long[n];
                hashes[0] = s.charAt(0);
                modifiers[0] = 1;
                for (int i=1; i<n; i++) {
                    hashes[i] = (hashes[i-1] * P + s.charAt(i)) % M;
                    modifiers[i] = (modifiers[i-1] * P) % M;
                }
                return new HashedString(hashes, modifiers);
            }

            /**
             * Indices are inclusive.
             */
            long getHash(HashedString hashedString, int startIndex, int endIndex) {
                long[] hashes = hashedString.hashes;
                long[] modifiers = hashedString.modifiers;
                long result = hashes[endIndex];
                if (startIndex > 0) result -= (hashes[startIndex-1] * modifiers[endIndex-startIndex+1]) % M;
                if (result < 0) result += M;
                return result;
            }

            // Less interesting methods below



            /**
             * Efficient for 2 input parameter strings in particular.
             */
            HashedString[] hashString(String first, String second) {
                HashedString[] array = new HashedString[2];
                int n = first.length();
                long[] modifiers = new long[n];
                modifiers[0] = 1;

                long[] firstHashes = new long[n];
                firstHashes[0] = first.charAt(0);
                array[0] = new HashedString(firstHashes, modifiers);

                long[] secondHashes = new long[n];
                secondHashes[0] = second.charAt(0);
                array[1] = new HashedString(secondHashes, modifiers);

                for (int i=1; i<n; i++) {
                    modifiers[i] = (modifiers[i-1] * P) % M;
                    firstHashes[i] = (firstHashes[i-1] * P + first.charAt(i)) % M;
                    secondHashes[i] = (secondHashes[i-1] * P + second.charAt(i)) % M;
                }
                return array;
            }

            /**
             * Efficient for 3+ strings
             * More efficient than multiple hashString calls IF strings are same length.
             */
            HashedString[] hashString(String... strings) {
                HashedString[] array = new HashedString[strings.length];
                int n = strings[0].length();
                long[] modifiers = new long[n];
                modifiers[0] = 1;
                for (int j=0; j<strings.length; j++) {
                    // if all strings are not same length, defer work to another method
                    if (strings[j].length() != n) {
                        for (int i=0; i<n; i++) {
                            array[i] = hashString(strings[i]);
                        }
                        return array;
                    }

                    // otherwise initialize stuff
                    long[] hashes = new long[n];
                    hashes[0] = strings[j].charAt(0);
                    array[j] = new HashedString(hashes, modifiers);
                }
                for (int i=1; i<n; i++) {
                    modifiers[i] = (modifiers[i-1] * P) % M;
                    for (int j=0; j<strings.length; j++) {
                        String s = strings[j];
                        long[] hashes = array[j].hashes;
                        hashes[i] = (hashes[i-1] * P + s.charAt(i)) % M;
                    }
                }
                return array;
            }

            void initializePandM() {
                ArrayList<Long> modOptions = new ArrayList<>(20);
                modOptions.add(353873237L);
                modOptions.add(353875897L);
                modOptions.add(353878703L);
                modOptions.add(353882671L);
                modOptions.add(353885303L);
                modOptions.add(353888377L);
                modOptions.add(353893457L);
                P = modOptions.get(new Random().nextInt(modOptions.size()));

                modOptions.clear();
                modOptions.add(452940277L);
                modOptions.add(452947687L);
                modOptions.add(464478431L);
                modOptions.add(468098221L);
                modOptions.add(470374601L);
                modOptions.add(472879717L);
                modOptions.add(472881973L);
                M = modOptions.get(new Random().nextInt(modOptions.size()));

            }
        }

        private static class Prob {

            /** For heavy calculations on probabilities, this class
             *  provides more accuracy & efficiency than doubles.
             *  Math explained: https://en.wikipedia.org/wiki/Log_probability
             *  Quick start:
             *      - Instantiate probabilities, eg. Prob a = new Prob(0.75)
             *      - add(), multiply() return new objects, can perform on nulls & NaNs.
             *      - get() returns probability as a readable double */

            /** Logarithmized probability. Note: 0% represented by logP NaN. */
            private double logP;

            /** Construct instance with real probability. */
            public Prob(double real) {
                if (real > 0) this.logP = Math.log(real);
                else this.logP = Double.NaN;
            }

            /** Construct instance with already logarithmized value. */
            static boolean dontLogAgain = true;
            public Prob(double logP, boolean anyBooleanHereToChooseThisConstructor) {
                this.logP = logP;
            }

            /** Returns real probability as a double. */
            public double get() {
                return Math.exp(logP);
            }

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

            /***************** STATIC METHODS BELOW ********************/

            /** Note: returns NaN only when a && b are both NaN/null. */
            public static Prob add(Prob a, Prob b) {
                if (nullOrNaN(a) && nullOrNaN(b)) return new Prob(Double.NaN, dontLogAgain);
                if (nullOrNaN(a)) return copy(b);
                if (nullOrNaN(b)) return copy(a);

                double x = Math.max(a.logP, b.logP);
                double y = Math.min(a.logP, b.logP);
                double sum = x + Math.log(1 + Math.exp(y - x));
                return new Prob(sum, dontLogAgain);
            }

            /** Note: multiplying by null or NaN produces NaN (repping 0% real prob). */
            public static Prob multiply(Prob a, Prob b) {
                if (nullOrNaN(a) || nullOrNaN(b)) return new Prob(Double.NaN, dontLogAgain);
                return new Prob(a.logP + b.logP, dontLogAgain);
            }

            /** Returns true if p is null or NaN. */
            private static boolean nullOrNaN(Prob p) {
                return (p == null || Double.isNaN(p.logP));
            }

            /** Returns a new instance with the same value as original. */
            private static Prob copy(Prob original) {
                return new Prob(original.logP, dontLogAgain);
            }
        }

        public class StronglyConnectedComponents {

            /** Kosaraju's algorithm */

            ArrayList<Integer>[] forw;
            ArrayList<Integer>[] bacw;

            /** Use: getCount(2, new int[] {1,2}, new int[] {2,1}) */
            public int getCount(int n, int[] mista, int[] minne) {
                forw = new ArrayList[n+1];
                bacw = new ArrayList[n+1];
                for (int i=1; i<=n; i++) {
                    forw[i] = new ArrayList<Integer>();
                    bacw[i] = new ArrayList<Integer>();
                }
                for (int i=0; i<mista.length; i++) {
                    int a = mista[i];
                    int b = minne[i];
                    forw[a].add(b);
                    bacw[b].add(a);
                }
                int count = 0;
                List<Integer> list = new ArrayList<Integer>();
                boolean[] visited = new boolean[n+1];
                for (int i=1; i<=n; i++) {
                    dfsForward(i, visited, list);
                }
                visited = new boolean[n+1];
                for (int i=n-1; i>=0; i--) {
                    int node = list.get(i);
                    if (visited[node]) continue;
                    count++;
                    dfsBackward(node, visited);
                }
                return count;
            }

            public void dfsForward(int i, boolean[] visited, List<Integer> list) {
                if (visited[i]) return;
                visited[i] = true;
                for (int neighbor : forw[i]) {
                    dfsForward(neighbor, visited, list);
                }
                list.add(i);
            }

            public void dfsBackward(int i, boolean[] visited) {
                if (visited[i]) return;
                visited[i] = true;
                for (int neighbor : bacw[i]) {
                    dfsBackward(neighbor, visited);
                }
            }
        }

        class DrawGrid {

            void draw(boolean[][] d) {
                System.out.print("  ");
                for (int x=0; x<d[0].length; x++) {
                    System.out.print(" " + x + " ");
                }
                System.out.println("");
                for (int y=0; y<d.length; y++) {
                    System.out.print(y + " ");
                    for (int x=0; x<d[0].length; x++) {
                        System.out.print((d[y][x] ? "[x]" : "[ ]"));
                    }
                    System.out.println("");
                }
            }

            void draw(int[][] d) {
                int max = 1;
                for (int y=0; y<d.length; y++) {
                    for (int x=0; x<d[0].length; x++) {
                        max = Math.max(max, ("" + d[y][x]).length());
                    }
                }
                System.out.print("  ");
                String format = "%" + (max+2) + "s";
                for (int x=0; x<d[0].length; x++) {
                    System.out.print(String.format(format, x) + " ");
                }
                format = "%" + (max) + "s";
                System.out.println("");
                for (int y=0; y<d.length; y++) {
                    System.out.print(y + " ");
                    for (int x=0; x<d[0].length; x++) {
                        System.out.print(" [" + String.format(format, (d[y][x])) + "]");
                    }
                    System.out.println("");
                }
            }

        }

        class BaseConverter {

            /* Palauttaa luvun esityksen kannassa base */
            public String convert(Long number, int base) {
                return Long.toString(number, base);
            }

            /* Palauttaa luvun esityksen kannassa baseTo, kun annetaan luku Stringinä kannassa baseFrom */
            public String convert(String number, int baseFrom, int baseTo) {
                return Long.toString(Long.parseLong(number, baseFrom), baseTo);
            }

            /* Tulkitsee kannassa base esitetyn luvun longiksi (kannassa 10) */
            public long longify(String number, int baseFrom) {
                return Long.parseLong(number, baseFrom);
            }
        }

        class Binary implements Comparable<Binary> {

            /**
             * Use example: Binary b = new Binary(Long.toBinaryString(53249834L));
             *
             * When manipulating small binary strings, instantiate new Binary(string)
             * When just reading large binary strings, instantiate new Binary(string,true)
             * get(int i) returns a character '1' or '0', not an int.
             */

            private boolean[] d;
            private int first; // Starting from left, the first (most remarkable) '1'
            public int length;


            public Binary(String binaryString) {
                this(binaryString, false);
            }
            public Binary(String binaryString, boolean initWithMinArraySize) {
                length = binaryString.length();
                int size = Math.max(2*length, 1);
                first = length/4;
                if (initWithMinArraySize) {
                    first = 0;
                    size = Math.max(length, 1);
                }
                d = new boolean[size];
                for (int i=0; i<length; i++) {
                    if (binaryString.charAt(i) == '1') d[i+first] = true;
                }
            }

            public void addFirst(char c) {
                if (first-1 < 0) doubleArraySize();
                first--;
                d[first] = (c == '1' ? true : false);
                length++;
            }

            public void addLast(char c) {
                if (first+length >= d.length) doubleArraySize();
                d[first+length] = (c == '1' ? true : false);
                length++;
            }

            private void doubleArraySize() {
                boolean[] bigArray = new boolean[(d.length+1) * 2];
                int newFirst = bigArray.length / 4;
                for (int i=0; i<length; i++) {
                    bigArray[i + newFirst] = d[i + first];
                }
                first = newFirst;
                d = bigArray;
            }

            public boolean flip(int i) {
                boolean value = (this.d[first+i] ? false : true);
                this.d[first+i] = value;
                return value;
            }

            public void set(int i, char c) {
                boolean value = (c == '1' ? true : false);
                this.d[first+i] = value;
            }

            public char get(int i) {
                return (this.d[first+i] ? '1' : '0');
            }

            @Override
            public int compareTo(Binary o) {
                if (this.length != o.length) return this.length - o.length;
                int len = this.length;
                for (int i=0; i<len; i++) {
                    int diff = this.get(i) - o.get(i);
                    if (diff != 0) return diff;
                }
                return 0;
            }

            @Override
            public String toString() {
                StringBuilder sb = new StringBuilder();
                for (int i=0; i<length; i++) {
                    sb.append(d[i+first] ? '1' : '0');
                }
                return sb.toString();
            }


        }

        class BinomialCoefficients {
            /** Total number of K sized unique combinations from pool of size N (unordered)
             N! / ( K! (N - K)! )   */

            /** For simple queries where output fits in long. */
            public long biCo(long n, long k) {
                long r = 1;
                if (k > n) return 0;
                for (long d = 1; d <= k; d++) {
                    r *= n--;
                    r /= d;
                }
                return r;
            }

            /** For multiple queries with same n, different k. */
            public long[] precalcBinomialCoefficientsK(int n, int maxK) {
                long v[] = new long[maxK+1];
                v[0] = 1; // nC0 == 1
                for (int i=1; i<=n; i++) {
                    for (int j=Math.min(i,maxK); j>0; j--) {
                        v[j] = v[j] + v[j-1]; // Pascal's triangle
                    }
                }
                return v;
            }

            /** When output needs % MOD. */
            public long[] precalcBinomialCoefficientsK(int n, int k, long M) {
                long v[] = new long[k+1];
                v[0] = 1; // nC0 == 1
                for (int i=1; i<=n; i++) {
                    for (int j=Math.min(i,k); j>0; j--) {
                        v[j] = v[j] + v[j-1]; // Pascal's triangle
                        v[j] %= M;
                    }
                }
                return v;
            }
        }

        class Trie {
            int N;
            int Z;
            int nextFreeId;
            int[][] pointers;
            boolean[] end;

            /** maxLenSum = maximum possible sum of length of words */
            public Trie(int maxLenSum, int alphabetSize) {
                this.N = maxLenSum;
                this.Z = alphabetSize;
                this.nextFreeId = 1;
                pointers = new int[N+1][alphabetSize];
                end = new boolean[N+1];
            }

            public void addWord(String word) {
                int curr = 0;
                for (int j=0; j<word.length(); j++) {
                    int c = word.charAt(j) - 'a';
                    int next = pointers[curr][c];
                    if (next == 0) {
                        next = nextFreeId++;
                        pointers[curr][c] = next;
                    }
                    curr = next;
                }
                int last = word.charAt(word.length()-1) - 'a';
                end[last] = true;
            }

            public boolean hasWord(String word) {
                int curr = 0;
                for (int j=0; j<word.length(); j++) {
                    int c = word.charAt(j) - 'a';
                    int next = pointers[curr][c];
                    if (next == 0) return false;
                    curr = next;
                }
                int last = word.charAt(word.length()-1) - 'a';
                return end[last];
            }

        }

        private class IO extends PrintWriter {
            private InputStreamReader r;
            private static final int BUFSIZE = 1 << 15;
            private char[] buf;
            private int bufc;
            private int bufi;
            private StringBuilder sb;

            public IO() {
                super(new BufferedOutputStream(System.out));
                r = new InputStreamReader(System.in);

                buf = new char[BUFSIZE];
                bufc = 0;
                bufi = 0;
                sb = new StringBuilder();
            }

            /** Print, flush, return nextInt. */
            private int queryInt(String s) {
                io.println(s);
                io.flush();
                return nextInt();
            }

            /** Print, flush, return nextLong. */
            private long queryLong(String s) {
                io.println(s);
                io.flush();
                return nextLong();
            }

            /** Print, flush, return next word. */
            private String queryNext(String s) {
                io.println(s);
                io.flush();
                return next();
            }

            private void fillBuf() throws IOException {
                bufi = 0;
                bufc = 0;
                while(bufc == 0) {
                    bufc = r.read(buf, 0, BUFSIZE);
                    if(bufc == -1) {
                        bufc = 0;
                        return;
                    }
                }
            }

            private boolean pumpBuf() throws IOException {
                if(bufi == bufc) {
                    fillBuf();
                }
                return bufc != 0;
            }

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

            private void eatDelimiters() throws IOException {
                while(true) {
                    if(bufi == bufc) {
                        fillBuf();
                        if(bufc == 0) throw new RuntimeException("IO: Out of input.");
                    }

                    if(!isDelimiter(buf[bufi])) break;
                    ++bufi;
                }
            }

            public String next() {
                try {
                    sb.setLength(0);

                    eatDelimiters();
                    int start = bufi;

                    while(true) {
                        if(bufi == bufc) {
                            sb.append(buf, start, bufi - start);
                            fillBuf();
                            start = 0;
                            if(bufc == 0) break;
                        }

                        if(isDelimiter(buf[bufi])) break;
                        ++bufi;
                    }

                    sb.append(buf, start, bufi - start);

                    return sb.toString();
                } catch(IOException e) {
                    throw new RuntimeException("IO.next: Caught IOException.");
                }
            }

            public int nextInt() {
                try {
                    int ret = 0;

                    eatDelimiters();

                    boolean positive = true;
                    if(buf[bufi] == '-') {
                        ++bufi;
                        if(!pumpBuf()) throw new RuntimeException("IO.nextInt: Invalid int.");
                        positive = false;
                    }

                    boolean first = true;
                    while(true) {
                        if(!pumpBuf()) break;
                        if(isDelimiter(buf[bufi])) {
                            if(first) throw new RuntimeException("IO.nextInt: Invalid int.");
                            break;
                        }
                        first = false;

                        if(buf[bufi] >= '0' && buf[bufi] <= '9') {
                            if(ret < -214748364) throw new RuntimeException("IO.nextInt: Invalid int.");
                            ret *= 10;
                            ret -= (int)(buf[bufi] - '0');
                            if(ret > 0) throw new RuntimeException("IO.nextInt: Invalid int.");
                        } else {
                            throw new RuntimeException("IO.nextInt: Invalid int.");
                        }

                        ++bufi;
                    }

                    if(positive) {
                        if(ret == -2147483648) throw new RuntimeException("IO.nextInt: Invalid int.");
                        ret = -ret;
                    }

                    return ret;
                } catch(IOException e) {
                    throw new RuntimeException("IO.nextInt: Caught IOException.");
                }
            }

            public long nextLong() {
                try {
                    long ret = 0;

                    eatDelimiters();

                    boolean positive = true;
                    if(buf[bufi] == '-') {
                        ++bufi;
                        if(!pumpBuf()) throw new RuntimeException("IO.nextLong: Invalid long.");
                        positive = false;
                    }

                    boolean first = true;
                    while(true) {
                        if(!pumpBuf()) break;
                        if(isDelimiter(buf[bufi])) {
                            if(first) throw new RuntimeException("IO.nextLong: Invalid long.");
                            break;
                        }
                        first = false;

                        if(buf[bufi] >= '0' && buf[bufi] <= '9') {
                            if(ret < -922337203685477580L) throw new RuntimeException("IO.nextLong: Invalid long.");
                            ret *= 10;
                            ret -= (long)(buf[bufi] - '0');
                            if(ret > 0) throw new RuntimeException("IO.nextLong: Invalid long.");
                        } else {
                            throw new RuntimeException("IO.nextLong: Invalid long.");
                        }

                        ++bufi;
                    }

                    if(positive) {
                        if(ret == -9223372036854775808L) throw new RuntimeException("IO.nextLong: Invalid long.");
                        ret = -ret;
                    }

                    return ret;
                } catch(IOException e) {
                    throw new RuntimeException("IO.nextLong: Caught IOException.");
                }
            }

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

        }

        void print(Object output) {
            io.println(output);
        }

        void done(Object output) {
            print(output);
            done();
        }

        void done() {
            io.close();
            throw new RuntimeException("Clean exit");
        }

        long min(long... v) {
            long ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.min(ans, v[i]);
            }
            return ans;
        }

        double min(double... v) {
            double ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.min(ans, v[i]);
            }
            return ans;
        }

        int min(int... v) {
            int ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.min(ans, v[i]);
            }
            return ans;
        }

        long max(long... v) {
            long ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.max(ans, v[i]);
            }
            return ans;
        }

        double max(double... v) {
            double ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.max(ans, v[i]);
            }
            return ans;
        }

        int max(int... v) {
            int ans = v[0];
            for (int i=1; i<v.length; i++) {
                ans = Math.max(ans, v[i]);
            }
            return ans;
        }

        void debug(Object output) {
            if (DEBUG_PRINTS) {
                System.out.println(output);
            }
        }
    }

}