Topology

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.

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.

Examples

Input

1 10

Output

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

Input

2 1000

Output

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

Input

2 1001

Output

Impossible

Input

1 11

Output

Possible 0 1 1

inputFormat

outputFormat

Output below.

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.

Examples

Input

1 10

Output

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

Input

2 1000

Output

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

Input

2 1001

Output

Impossible

Input

1 11

Output

Possible 0 1 1

样例

2
1000

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

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