Problem:
Score : 900 points

Problem Statement

AtCoDeer the deer wants a directed graph that satisfies the following conditions:

The number of vertices, N, is at most 300.

There must not be self-loops or multiple edges.

The vertices are numbered from 1 through N.

Each edge has either an integer weight between 0 and 100 (inclusive), or a label X or Y.

For every pair of two integers (x,y) such that 1 ≤ x ≤ A, 1 ≤ y ≤ B,
the shortest distance from Vertex S to Vertex T in the graph where the edges labeled X have the weight x and the edges labeled Y have the weight y, is d_{x,y}.

Construct such a graph (and a pair of S and T) for him, or report that it does not exist.
Refer to Output section for output format.

Constraints

1 ≤ A,B ≤ 10

1 ≤ d_{x,y} ≤ 100 (1 ≤ x ≤ A, 1 ≤ y ≤ B)

All input values are integers.

Input

Input is given from Standard Input in the following format:

A B
d_{1,1} d_{1,2} .. d_{1,B}
d_{2,1} d_{2,2} .. d_{2,B}
:
d_{A,1} d_{A,2} .. d_{A,B}

Output

If no graph satisfies the condition, print Impossible.

If there exists a graph that satisfies the condition, print Possible in the first line.
Then, in the subsequent lines, print the constructed graph in the following format:

N M
u_1 v_1 c_1
u_2 v_2 c_2
:
u_M v_M c_M
S T

Here, M is the number of the edges, and u_i, v_i, c_i represent edges as follows: there is an edge from Vertex u_i to Vertex v_i whose weight or label is c_i.

Also refer to Sample Outputs.

Sample Input 1

2 3
1 2 2
1 2 3

Sample Output 1

Possible
3 4
1 2 X
2 3 1
3 2 Y
1 3 Y
1 3

Sample Input 2

1 3
100 50 1

Sample Output 2

Impossible

Solution in Java (OpenJDK 11.0.6) (judge verdict: Runtime Error)

import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.lang.reflect.Field;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.NoSuchElementException;
import java.util.Objects;
import java.util.TreeSet;
import java.util.function.IntUnaryOperator;
import java.util.function.LongUnaryOperator;


public class Main {
    public static void main(String[] args) throws Exception {
        Field f = System.out.getClass().getDeclaredField("autoFlush");
        f.setAccessible(true);
        f.set(System.out, false);
        execute(System.in, System.out);
    }

    public static void execute(InputStream in, OutputStream out) {
        ExtendedScanner s = new ExtendedScanner(in);
        Out o = new Out(out);
        solve(s, o);
        o.flush();
        o.close();
    }

    public static void solve(ExtendedScanner sc, Out out) {
        int a = sc.nextInt();
        int b = sc.nextInt();
        int[][] d = sc.ints(a, b);
        boolean ng = false;
        for (int x = 0; x < a; x++) {
            for (int y = 1; y < b - 1; y++) {
                int dy1 = d[x][y] - d[x][y - 1];
                int dy2 = d[x][y + 1] - d[x][y];
                ng |= dy1 < 0 || dy2 < 0 || dy1 < dy2;
            }
        }
        for (int x = 1; x < a - 1; x++) {
            for (int y = 0; y < b; y++) {
                int dx1 = d[x][y] - d[x - 1][y];
                int dx2 = d[x + 1][y] - d[x][y];
                ng |= dx1 < 0 || dx2 < 0 || dx1 < dx2;
            }
        }
        if (ng) {
            out.writeln("Impossible");
            return;
        }
        var planes = new TreeSet<IntPair3>((p1, p2) -> {
            if (p1.fst == p2.fst) {
                if (p1.snd == p2.snd) {
                    return p1.trd - p2.trd;
                } else {
                    return p1.snd - p2.snd;
                }
            } else {
                return p1.fst - p2.fst;
            }
        });
        if (a == 1) {
            for (int y = 0; y < b - 1; y++) {
                planes.add(new IntPair3(0, d[0][y + 1] - d[0][y], d[0][y]));
            }
        } else if (b == 1) {
            for (int x = 0; x < a - 1; x++) {
                planes.add(new IntPair3(d[x + 1][0] - d[x][0], 0, d[x][0]));
            }
        } else {
            int cnt = 0;
            boolean[][] ok = new boolean[a][b];
            {
                int dx = d[1][0] - d[0][0];
                int dy = d[0][1] - d[0][0];
                int c = d[0][0] - dx - dy;
                ng |= c < 0;
                var pl = new IntPair3(dx, dy, c);
                planes.add(pl);
                for (int xx = 0; xx < a; xx++) {
                    for (int yy = 0; yy < b; yy++) {
                        if ((xx + 1) * pl.fst + (yy + 1) * pl.snd + pl.trd == d[xx][yy]) {
                            ok[xx][yy] = true;
                            cnt++;
                        }
                    }
                }
            }
            int[] dx8 = new int[]{0, 1, -1, 0, 1, -1, 1, -1};
            int[] dy8 = new int[]{1, 0, 0, -1, 1, -1, -1, 1};
            while (cnt < a * b) {
                for (int x = 0; x < a; x++) {
                    for (int y = 0; y < b; y++) {
                        if (ok[x][y]) continue;
                        var p0 = new IntPair3(x + 1, y + 1, d[x][y]);
                        IntPair3 p1 = null, p2 = null;
                        for (int dr = 0; dr < 8; dr++) {
                            int nx = x + dx8[dr];
                            int ny = y + dy8[dr];
                            if (nx < 0 || nx >= a || ny < 0 || ny >= b || !ok[nx][ny]) continue;
                            if (p1 == null) {
                                p1 = new IntPair3(nx + 1, ny + 1, d[nx][ny]);
                            } else {
                                p2 = new IntPair3(nx + 1, ny + 1, d[nx][ny]);
                                break;
                            }
                        }
                        if (p2 == null) continue;
                        var pl = plane(p0, p1, p2);
                        ng |= pl.trd < 0;
                        planes.add(pl);
                        for (int xx = 0; xx < a; xx++) {
                            for (int yy = 0; yy < b; yy++) {
                                if (ok[xx][yy]) continue;
                                if ((xx + 1) * pl.fst + (yy + 1) * pl.snd + pl.trd == d[xx][yy]) {
                                    ok[xx][yy] = true;
                                    cnt++;
                                }
                            }
                        }
                    }
                }
            }
        }
        if (ng) {
            out.writeln("Impossible");
            return;
        }
        var edges = new ArrayList<Edge>();
        int i = 2;
        for (var p : planes) {
            int xe = p.fst;
            int ye = p.snd;
            int ce = p.trd;
            edges.add(new Edge(0, i, ce));
            for (int x = 0; x < xe; x++) {
                edges.add(new Edge(i++, i, "X"));
            }
            for (int y = 0; y < ye; y++) {
                edges.add(new Edge(i++, i, "Y"));
            }
            edges.add(new Edge(i++, 1, 0));
        }
        out.writeln("Possible");
        out.write(i).writeSpace().writeln(edges.size());
        edges.forEach(out::writeln);
        out.write(1).writeSpace().writeln(2);
    }

    public static IntPair3 plane(IntPair3 a, IntPair3 b, IntPair3 c) {
        b.fst -= a.fst; b.snd -= a.snd; b.trd -= a.trd;
        c.fst -= a.fst; c.snd -= a.snd; c.trd -= a.trd;
        System.err.println(a);
        System.err.println(b);
        System.err.println(c);
        var op = outerProduct(b, c);
        System.err.println(op);
        int ca = op.fst;
        int cb = op.snd;
        int cc = op.trd;
        int d = -(ca * a.fst + cb * a.snd + cc * a.trd);
        if (((ca % cc) | (cb % cc) | (d % cc)) != 0) {
            throw new IllegalArgumentException();
        }
        ca /= -cc; cb /= -cc; d /= -cc;
        b.fst += a.fst; b.snd += a.snd; b.trd += a.trd;
        c.fst += a.fst; c.snd += a.snd; c.trd += a.trd;
        var plane = new IntPair3(ca, cb, d);
        return plane;
    }

    public static IntPair3 outerProduct(IntPair3 a, IntPair3 b) {
        int x = a.snd * b.trd - a.trd * b.snd;
        int y = a.trd * b.fst - a.fst * b.trd;
        int z = a.fst * b.snd - a.snd * b.fst;
        return new IntPair3(x, y, z);
    }
}

class Edge {
    int from;
    int to;
    Object cost;
    Edge(int from, int to, Object cost) {
        this.from = from + 1;
        this.to = to + 1;
        this.cost = cost;
    }
    public String toString() {
        return from + " " + to + " " + cost;
    }
}


/**
 * @author https://atcoder.jp/users/suisen
 */
class BasicScanner {
    private final InputStream in;
    private final byte[] buf = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;
    public BasicScanner(InputStream in) {this.in = in;}
    private boolean hasNextByte() {
        if (ptr < buflen) return true;
        ptr = 0;
        try {buflen = in.read(buf);}
        catch (final IOException e) {e.printStackTrace();}
        return buflen > 0;
    }
    private int readByte() {return hasNextByte() ? buf[ptr++] : -1;}
    public boolean hasNext() {
        while (hasNextByte() && !(33 <= buf[ptr] && buf[ptr] <= 126)) ptr++;
        return hasNextByte();
    }
    private StringBuilder nextSequence() {
        if (!hasNext()) throw new NoSuchElementException();
        final StringBuilder sb = new StringBuilder();
        int b = readByte();
        while (33 <= b && b <= 126) {sb.appendCodePoint(b); b = readByte();}
        return sb;
    }
    public char nextChar() {
        return (char) readByte();
    }
    public String next() {
        return nextSequence().toString();
    }
    public String next(int len) {
        return new String(nextChars(len));
    }
    public char[] nextChars() {
        final StringBuilder sb = nextSequence();
        int l = sb.length();
        char[] dst = new char[l];
        sb.getChars(0, l, dst, 0);
        return dst;
    }
    public char[] nextChars(int len) {
        if (!hasNext()) throw new NoSuchElementException();
        char[] s = new char[len];
        int i = 0;
        int b = readByte();
        while (33 <= b && b <= 126 && i < len) {s[i++] = (char) b; b = readByte();}
        if (i != len) throw new NoSuchElementException(String.format("length %d is longer than the sequence.", len));
        return s;
    }
    public long nextLong() {
        if (!hasNext()) throw new NoSuchElementException();
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {minus = true; b = readByte();}
        if (b < '0' || '9' < b) throw new NumberFormatException();
        while (true) {
            if ('0' <= b && b <= '9') n = n * 10 + b - '0';
            else if (b == -1 || !(33 <= b && b <= 126)) return minus ? -n : n;
            else throw new NumberFormatException();
            b = readByte();
        }
    }
    public int nextInt() {return Math.toIntExact(nextLong());}
    public double nextDouble() {return Double.parseDouble(next());}
}


/**
 * @author https://atcoder.jp/users/suisen
 */
class Out {
    private final OutputStream out;
    private byte[] buf = new byte[1024];
    private int ptr = 0;
    private static final int AUTO_FLUSH_THRETHOLD = 1 << 17;

    public Out(OutputStream out) {
        this.out = out;
    }

    public void flush() {
        try {
            out.write(buf, 0, ptr);
            out.flush();
            ptr = 0;
        } catch (IOException e) {e.printStackTrace();}
    }

    public void close() {
        try {out.close();} catch (IOException e) {e.printStackTrace();}
    }

    public Out writeln() {return write('\n');}
    public Out writeln(Object o) {return write(o).write('\n');}
    public Out writeln(String s) {return write(s).write('\n');}
    public Out writeln(char[] c) {return write(c).write('\n');}
    public Out writeln(char   c) {return write(c).write('\n');}
    public Out writeln(byte   b) {return write(b).write('\n');}
    public Out writeln(int    x) {return write(x).write('\n');}
    public Out writeln(long   x) {return write(x).write('\n');}
    public Out writeln(double d) {return write(d).write('\n');}

    public Out writeSpace() {return write(' ');}
    
    public Out write(Object o) {
        return write(o.toString());
    }

    public Out write(String s) {
        try {
            Field strValueField = s.getClass().getDeclaredField("value");
            strValueField.setAccessible(true);
            byte[] b = (byte[]) strValueField.get(s);
            int l = s.length();
            if (l > AUTO_FLUSH_THRETHOLD) {
                flush();
                int i = 0;
                while (i + AUTO_FLUSH_THRETHOLD < l) {
                    out.write(b, i, AUTO_FLUSH_THRETHOLD);
                    out.flush();
                    i += AUTO_FLUSH_THRETHOLD;
                }
                ensureCapacity(l - i);
                System.arraycopy(b, i, buf, 0, l - i);
                ptr = l - i;
            } else {
                ensureCapacity(ptr + l);
                System.arraycopy(b, 0, buf, ptr, l);
                ptr += l;
            }
        } catch (Exception e) {e.printStackTrace();}
        return this;
    }

    public Out write(char[] c) {
        int l = c.length;
        if (l > AUTO_FLUSH_THRETHOLD) {
            flush();
            ensureCapacity(AUTO_FLUSH_THRETHOLD);
            int i = 0;
            while (i + AUTO_FLUSH_THRETHOLD < l) {
                for (int j = 0; j < AUTO_FLUSH_THRETHOLD; j++) {
                    buf[ptr++] = (byte) c[i + j];
                }
                flush();
                i += AUTO_FLUSH_THRETHOLD;
            }
            for (; i < l; i++) {
                buf[ptr++] = (byte) c[i];
            }
        } else {
            ensureCapacity(ptr + l);
            for (char ch : c) buf[ptr++] = (byte) ch;
        }
        return this;
    }

    public Out write(char c) {
        ensureCapacity(ptr + 1);
        buf[ptr++] = (byte) c;
        return this;
    }

    public Out write(byte b) {
        ensureCapacity(ptr + 1);
        buf[ptr++] = b;
        return this;
    }

    public Out write(int x) {
        if (x == 0) {
            ensureCapacity(ptr + 1);
            buf[ptr++] = '0';
            return this;
        }
        int d = stringSize(x);
        ensureCapacity(ptr + d);
        if (x < 0) {
            buf[ptr++] = '-';
            x = -x;
            d--;
        }
        int j = ptr + d; 
        while (x > 0) {
            buf[--j] = (byte) ('0' + (x % 10));
            x /= 10;
        }
        ptr += d;
        return this;
    }

    public Out write(long x) {
        if (x == 0) {
            ensureCapacity(ptr + 1);
            buf[ptr++] = '0';
            return this;
        }
        int d = stringSize(x);
        ensureCapacity(ptr + d);
        if (x < 0) {
            buf[ptr++] = '-';
            x = -x;
            d--;
        }
        int j = ptr + d; 
        while (x > 0) {
            buf[--j] = (byte) ('0' + (x % 10));
            x /= 10;
        }
        ptr += d;
        return this;
    }

    public Out write(double d) {
        return write(Double.toString(d));
    }

    private void ensureCapacity(int cap) {
        if (cap > AUTO_FLUSH_THRETHOLD) {
            flush();
        }
        if (cap >= buf.length) {
            int newLength = buf.length;
            while (newLength < cap) newLength <<= 1;
            byte[] newBuf = new byte[newLength];
            System.arraycopy(buf, 0, newBuf, 0, buf.length);
            buf = newBuf;
        }
    }
    private static int stringSize(long x) {
        int d = 1;
        if (x >= 0) {d = 0; x = -x;}
        long p = -10;
        for (int i = 1; i < 19; i++, p *= 10) if (x > p) return i + d;
        return 19 + d;
    }
    private static int stringSize(int x) {
        int d = 1;
        if (x >= 0) {d = 0; x = -x;}
        int p = -10;
        for (int i = 1; i < 10; i++, p *= 10) if (x > p) return i + d;
        return 10 + d;
    }
}


/**
 * @author https://atcoder.jp/users/suisen
 */
final class ExtendedScanner extends BasicScanner {
    public ExtendedScanner(InputStream in) {super(in);}
    public int[] ints(final int n) {
        final int[] a = new int[n];
        Arrays.setAll(a, $ -> nextInt());
        return a;
    }
    public int[] ints(final int n, final IntUnaryOperator f) {
        final int[] a = new int[n];
        Arrays.setAll(a, $ -> f.applyAsInt(nextInt()));
        return a;
    }
    public int[][] ints(final int n, final int m) {
        final int[][] a = new int[n][];
        Arrays.setAll(a, $ -> ints(m));
        return a;
    }
    public int[][] ints(final int n, final int m, final IntUnaryOperator f) {
        final int[][] a = new int[n][];
        Arrays.setAll(a, $ -> ints(m, f));
        return a;
    }
    public long[] longs(final int n) {
        final long[] a = new long[n];
        Arrays.setAll(a, $ -> nextLong());
        return a;
    }
    public long[] longs(final int n, final LongUnaryOperator f) {
        final long[] a = new long[n];
        Arrays.setAll(a, $ -> f.applyAsLong(nextLong()));
        return a;
    }
    public long[][] longs(final int n, final int m) {
        final long[][] a = new long[n][];
        Arrays.setAll(a, $ -> longs(m));
        return a;
    }
    public long[][] longs(final int n, final int m, final LongUnaryOperator f) {
        final long[][] a = new long[n][];
        Arrays.setAll(a, $ -> longs(m, f));
        return a;
    }
    public char[][] charArrays(final int n) {
        final char[][] c = new char[n][];
        Arrays.setAll(c, $ -> nextChars());
        return c;
    }
    public char[][] charArrays(final int n, final int m) {
        final char[][] c = new char[n][];
        Arrays.setAll(c, $ -> nextChars(m));
        return c;
    }
    public double[] doubles(final int n) {
        final double[] a = new double[n];
        Arrays.setAll(a, $ -> nextDouble());
        return a;
    }
    public double[][] doubles(final int n, final int m) {
        final double[][] a = new double[n][];
        Arrays.setAll(a, $ -> doubles(m));
        return a;
    }
    public String[] strings(final int n) {
        final String[] s = new String[n];
        Arrays.setAll(s, $ -> next());
        return s;
    }
    public String[] strings(final int n, final int m) {
        final String[] s = new String[n];
        Arrays.setAll(s, $ -> next(m));
        return s;
    }
}


/**
 * @author https://atcoder.jp/users/suisen
 */
final class IntPair3 {
    public int fst, snd, trd;
    public IntPair3(final int fst, final int snd, final int trd) {this.fst = fst; this.snd = snd; this.trd = trd;}
    @Override @SuppressWarnings("all")
    public boolean equals(final Object o) {
        if (this == o) return true;
        if (!(o instanceof IntPair3)) return false;
        final IntPair3 p = (IntPair3) o;
        return this.fst == p.fst && this.snd == p.snd && this.trd == p.trd;
    }
    @Override
    public int hashCode() {return Objects.hash(this.fst, this.snd, this.trd);}
    @Override
    public String toString() {return "(" + this.fst + ", " + this.snd + ", " + this.trd + ")";}
}