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 C++ (GCC 9.2.1) (judge verdict: Wrong Answer)

#include<bits/stdc++.h>
#define pb push_back
#define all(x) x.begin(), x.end()
using namespace std;
const int maxn = 303, M = 100;
int a, b, d[14][14], C[M+3][M+3], sz = 1;
signed main() {
	cin.tie(0)->sync_with_stdio(0);
	cin >> a >> b;
	for(int i = 1; i <= a; i++)
		for(int j = 1; j <= b; j++)
			cin >> d[i][j];
	for(int as = 0; as <= M; as++) {
		for(int bs = 0; bs <= M; bs++) {
			int mn = 0;
			for(int i =1; i <= a; i++)
				for(int j = 1; j <= b; j++)
					mn = max(mn, d[i][j] - as*i - bs*j);
			C[as][bs] = mn;
		}
	}
	for(int i = 1; i <= a; i++)
		for(int j = 1; j <= b; j++) {
			int cst = 1<<30;
			for(int as = 0; as <= M; as++)
				for(int bs = 0; bs <= M; bs++)
					cst = min(cst, as*i + bs*j + C[as][bs]);
			if(cst != d[i][j]) return cout << "Impossible\n", 0;
		}
	
	cout << "Possible\n";
	cout << 2*M+2 << " " << (M+1)*(M+1) + 2*M << '\n';
	for(int i = 1; i <= M; i++)
		cout << sz << " " << sz+1 << " X\n", sz++;
	sz++;
	for(int i = 1; i <= M; i++)
		cout << sz << " " << sz+1 << " Y\n", sz++;
	for(int i = 0; i <= M; i++)
		for(int j = 0; j <= M; j++)
			cout << 1+i << " " << M+2+j << " " << C[i][j] << '\n';
}