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Contest: Task: Related: TaskD TaskF

Score : $1400$ points

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.

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