Problem:
Score : 1400 points

Problem Statement

Given are a positive integer N and a sequence of length 2^N consisting of 0s and 1s: A_0,A_1,\ldots,A_{2^N-1}.
Determine whether there exists a closed curve C that satisfies the condition below for all 2^N sets S \subseteq \{0,1,\ldots,N-1 \}. If the answer is yes, construct one such closed curve.

Let x = \sum_{i \in S} 2^i and B_S be the set of points \{ (i+0.5,0.5) | i \in S \}.

If there is a way to continuously move the closed curve C without touching B_S so that every point on the closed curve has a negative y-coordinate, A_x = 1.

If there is no such way, A_x = 0.

For instruction on printing a closed curve, see Output below.

Notes

C is said to be a closed curve if and only if:

C is a continuous function from [0,1] to \mathbb{R}^2 such that C(0) = C(1).

We say that a closed curve C can be continuously moved without touching a set of points X so that it becomes a closed curve D if and only if:

There exists a function f : [0,1] \times [0,1] \rightarrow \mathbb{R}^2 that satisfies all of the following.

f is continuous.

f(0,t) = C(t).

f(1,t) = D(t).

f(x,t) \notin X.

Constraints

1 \leq N \leq 8

A_i = 0,1 \quad (0 \leq i \leq 2^N-1)

A_0 = 1

Input

Input is given from Standard Input in the following format:

N
A_0A_1 \cdots A_{2^N-1}

Output

If there is no closed curve that satisfies the condition, print Impossible.

If such a closed curve exists, print Possible in the first line.
Then, print one such curve in the following format:

L
x_0 y_0
x_1 y_1
:
x_L y_L

This represents the closed polyline that passes (x_0,y_0),(x_1,y_1),\ldots,(x_L,y_L) in this order.

Here, all of the following must be satisfied:

0 \leq x_i \leq N, 0 \leq y_i \leq 1, and x_i, y_i are integers. (0 \leq i \leq L)

|x_i-x_{i+1}| + |y_i-y_{i+1}| = 1. (0 \leq i \leq L-1)

(x_0,y_0) = (x_L,y_L).

Additionally, the length of the closed curve L must satisfy 0 \leq L \leq 250000.

It can be proved that, if there is a closed curve that satisfies the condition in Problem Statement, there is also a closed curve that can be expressed in this format.

Sample Input 1

1
10

Sample Output 1

Possible
4
0 0
0 1
1 1
1 0
0 0

When S = \emptyset, we can move this curve so that every point on it has a negative y-coordinate.

When S = \{0\}, we cannot do so.

Sample Input 2

2
1000

Sample Output 2

Possible
6
1 0
2 0
2 1
1 1
0 1
0 0
1 0

The output represents the following curve:

Sample Input 3

2
1001

Sample Output 3

Impossible

Sample Input 4

1
11

Sample Output 4

Possible
0
1 1

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

typedef long long ll;
typedef pair<int,int> P;

#define fr first
#define sc second
#define chmax(a,b) a=max(a,b)
#define chmin(a,b) a=min(a,b)

vector<P> ret;

void pos(int i){
	for(int j=0;j<i;j++){
		ret.push_back(P(j+1,0));
	}
	ret.push_back(P(i,1));
	ret.push_back(P(i+1,1));
	ret.push_back(P(i+1,0));
	ret.push_back(P(i,0));
	for(int j=i-1;j>=0;j--){
		ret.push_back(P(j,0));
	}
}
void neg(int i){
	for(int j=0;j<i;j++){
		ret.push_back(P(j+1,0));
	}
	ret.push_back(P(i+1,0));
	ret.push_back(P(i+1,1));
	ret.push_back(P(i,1));
	ret.push_back(P(i,0));
	for(int j=i-1;j>=0;j--){
		ret.push_back(P(j,0));
	}
}

vector<P> f(vector<P> vec,int t){
	int cnt[10];
	for(int i=0;i<vec.size();i++){
		cnt[vec[i].sc]++;
	}
	int MIN=100;
	int id=-1;
	for(int i=0;i<MIN;i++){
		if(MIN>cnt[i]){
			MIN=cnt[i];
			id=i;
		}
	}
	vector<P> ret;
	for(int i=0;i<vec.size();i++){
		if(vec[i].sc!=id)ret.push_back(vec[i]);
		else {
			if(vec[i].fr==1){
				ret.push_back(P(1,id));
				ret.push_back(P(1,t));
				ret.push_back(P(-1,id));
				ret.push_back(P(-1,t));
			}
			else {
				ret.push_back(P(1,t));
				ret.push_back(P(1,id));
				ret.push_back(P(-1,t));
				ret.push_back(P(-1,id));
			}
		}
	}
	return ret;
}
	

int main(){
	int n;
	string s;
	cin>>n>>s;
	
	int a[300];
	for(int i=0;i<(1<<n);i++)a[i]=s[i]-'0';
	
	bool used[300];
	memset(used,0,sizeof used);
	
	int b[300];
	for(int i=0;i<300;i++)b[i]=1;
	
	for(int x=1;x<(1<<n);x++){
		used[x]=true;
		//if(used[x])cerr<<x<<endl;
		for(int y=0;y<x;y++){
			if((x&y)==y)if(used[y])used[x]=false;
		}
		//if(used[x])cerr<<x<<endl;
		for(int y=0;y<(1<<n);y++){
			if((x&y)==x)if(a[y]==1)used[x]=false;
		}
		//if(used[x])cerr<<x<<endl;
		if(used[x]){
			for(int y=0;y<(1<<n);y++){
				if((x&y)==x)b[y]=0;
			}
		}
	}
	
	bool ok=true;
	for(int i=0;i<(1<<n);i++)ok&=a[i]==b[i];
	if(!ok){
		puts("Impossible");
		return 0;
	}
	ret.push_back(P(0,0));
	for(int x=0;x<(1<<n);x++){
		if(!used[x])continue;
		int cnt=0;
		int c[10];
		for(int i=0;i<n;i++){
			if((x>>i)&1)c[cnt++]=i;
		}
		
		vector<P> vec;
		vec.push_back(P(1,c[0]));
		for(int i=1;i<cnt;i++)vec=f(vec,c[i]);
		for(int i=0;i<vec.size();i++){
			if(vec[i].fr==1)pos(vec[i].sc);
			else neg(vec[i].sc);
		}
		
		/*if(cnt==1){
			pos(c[0]);
		}
		else if(cnt==2){
			pos(c[0]);
			pos(c[1]);
			neg(c[0]);
			neg(c[1]);
		}
		else if(cnt==3){
			pos(c[0]);
			pos(c[1]);
			neg(c[0]);
			pos(c[2]);
			pos(c[0]);
			neg(c[1]);
			neg(c[0]);
			neg(c[2]);
		}
		else if(cnt==4){
			pos(c[0]);
			pos(c[1]);
			neg(c[0]);
			pos(c[2]);
			pos(c[3]);
			neg(c[2]);
			neg(c[3]);
			pos(c[0]);
			neg(c[1]);
			neg(c[0]);
			pos(c[3]);
			pos(c[2]);
			neg(c[3]);
			neg(c[2]);
		}
		else if(cnt==5){
			pos(c[0]);
			pos(c[1]);
			pos(c[4]);
			neg(c[1]);
			neg(c[4]);
			neg(c[0]);
			pos(c[2]);
			pos(c[3]);
			neg(c[2]);
			neg(c[3]);
			pos(c[0]);
			pos(c[4]);
			pos(c[1]);
			neg(c[4]);
			neg(c[1]);
			neg(c[0]);
			neg(c[3]);
			neg(c[2]);
			pos(c[3]);
			pos(c[2]);
		}
		else if(cnt==6){
			pos(c[0]);
			pos(c[5]);
			neg(c[0]);
			neg(c[5]);
			pos(c[1]);
			pos(c[4]);
			neg(c[1]);
			neg(c[4]);
			pos(c[5]);
			pos(c[0]);
			neg(c[5]);
			neg(c[0]);
			pos(c[2]);
			pos(c[3]);
			neg(c[2]);
			neg(c[3]);
			pos(c[0]);
			pos(c[5]);
			neg(c[0]);
			neg(c[5]);
			pos(c[4]);
			pos(c[1]);
			neg(c[4]);
			neg(c[1]);
			pos(c[5]);
			pos(c[0]);
			neg(c[5]);
			neg(c[0]);
			neg(c[3]);
			neg(c[2]);
			pos(c[3]);
			pos(c[2]);
		}
		else if(cnt==7){
			pos(c[0]);
			pos(c[5]);
			neg(c[0]);
			neg(c[5]);
			pos(c[1]);
			pos(c[6]);
			neg(c[1]);
			neg(c[6]);
			pos(c[4]);
			pos(c[6]);
			pos(c[1]);
			neg(c[6]);
			neg(c[1]);
			neg(c[4]);
			pos(c[5]);
			pos(c[0]);
			neg(c[5]);
			neg(c[0]);
			pos(c[2]);
			pos(c[3]);
			neg(c[2]);
			neg(c[3]);
			pos(c[0]);
			pos(c[5]);
			neg(c[0]);
			neg(c[5]);
			pos(c[4]);
			pos(c[1]);
			pos(c[6]);
			neg(c[1]);
			neg(c[6]);
			neg(c[4]);
			pos(c[6]);
			pos(c[1]);
			neg(c[6]);
			neg(c[1]);
			pos(c[5]);
			pos(c[0]);
			neg(c[5]);
			neg(c[0]);
			neg(c[3]);
			neg(c[2]);
			pos(c[3]);
			pos(c[2]);
		}
		else if(cnt==7){
			pos(c[0]);
			pos(c[5]);
			neg(c[0]);
			neg(c[5]);
			pos(c[1]);
			pos(c[6]);
			neg(c[1]);
			neg(c[6]);
			pos(c[4]);
			pos(c[7]);
			neg(c[4]);
			neg(c[7]);
			pos(c[6]);
			pos(c[1]);
			neg(c[6]);
			neg(c[1]);
			pos(c[7]);
			pos(c[4]);
			neg(c[7]);
			neg(c[4]);
			pos(c[5]);
			pos(c[0]);
			neg(c[5]);
			neg(c[0]);
			pos(c[2]);
			pos(c[3]);
			neg(c[2]);
			neg(c[3]);
			pos(c[0]);
			pos(c[5]);
			neg(c[0]);
			neg(c[5]);
			pos(c[4]);
			pos(c[7]);
			neg(c[4]);
			neg(c[7]);
			pos(c[1]);
			pos(c[6]);
			neg(c[1]);
			neg(c[6]);
			pos(c[7]);
			pos(c[4]);
			neg(c[7]);
			neg(c[4]);
			pos(c[6]);
			pos(c[1]);
			neg(c[6]);
			neg(c[1]);
			pos(c[5]);
			pos(c[0]);
			neg(c[5]);
			neg(c[0]);
			neg(c[3]);
			neg(c[2]);
			pos(c[3]);
			pos(c[2]);
		}*/
	}
	
	/*for(int i=0;i<n;i++){
		if(used[i][i]){
			for(int j=0;j<i;j++){
				ret.push_back(P(j+1,0));
			}
			ret.push_back(P(i,1));
			ret.push_back(P(i+1,1));
			ret.push_back(P(i+1,0));
			ret.push_back(P(i,0));
			for(int j=i-1;j>=0;j--){
				ret.push_back(P(j,0));
			}
		}
	}
	for(int i=0;i<n;i++){
		for(int k=i+1;k<n;k++){
			if(used[i][k]){
				for(int j=0;j<i;j++){
					ret.push_back(P(j+1,0));
				}
				ret.push_back(P(i,1));
				ret.push_back(P(i+1,1));
				ret.push_back(P(i+1,0));
				ret.push_back(P(i,0));
				for(int j=i-1;j>=0;j--){
					ret.push_back(P(j,0));
				}
				for(int j=0;j<k;j++){
					ret.push_back(P(j+1,0));
				}
				ret.push_back(P(k,1));
				ret.push_back(P(k+1,1));
				ret.push_back(P(k+1,0));
				ret.push_back(P(k,0));
				for(int j=k-1;j>=0;j--){
					ret.push_back(P(j,0));
				}
				for(int j=0;j<i;j++){
					ret.push_back(P(j+1,0));
				}
				ret.push_back(P(i+1,0));
				ret.push_back(P(i+1,1));
				ret.push_back(P(i,1));
				ret.push_back(P(i,0));
				for(int j=i-1;j>=0;j--){
					ret.push_back(P(j,0));
				}
				for(int j=0;j<k;j++){
					ret.push_back(P(j+1,0));
				}
				ret.push_back(P(k+1,0));
				ret.push_back(P(k+1,1));
				ret.push_back(P(k,1));
				ret.push_back(P(k,0));
				for(int j=k-1;j>=0;j--){
					ret.push_back(P(j,0));
				}
			}
		}
	}*/
	
	assert(ret.size()<=250000);
	
	puts("Possible");
	printf("%d\n",(int)(ret.size())-1);
	for(int i=0;i<ret.size();i++){
		printf("%d %d\n",ret[i].fr,ret[i].sc);
	}
}