Problem Statement

Given are a positive integer $N$ and a sequence of length $2^N$ consisting of $0$s and $1$s: $A_0,A_1,\ldots,A_{2^N-1}$. 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.

Note

$C$ 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.