You be The Judge, Again

Time limit2sMemory limit2048 MB

Summary

Given a 2^n by 2^n grid of color labels, check whether it is a valid tiling by L-shaped triominoes with exactly one empty cell.

Level: Medium4 of 10
Topics: Implementation, Simulation, Array, Brute force
Solved: No attempts yet

Problem

You are a judge, again! The contest you're judging includes the following problem:

You have one L-shaped triomino of each of $\frac{4^n-1}{3}$ different colors. Tile a $2^n$ by $2^n$ grid using each of these triominos such that there is exactly one blank square and all other squares are covered by exactly one square of such a triomino. All triominos must be used.

Your team is to write a checker for this problem. Validation of the input values and format has already taken place. You will be given a purported tiling of a $2^n$ by $2^n$ grid, where each square in the grid is either 0 or a positive integer from $1$ to $\frac{4^n-1}{3}$ representing one of the colors. Determine if it is, indeed, a covering of the grid with $\frac{4^n-1}{3}$ unique triominos and a single empty space.

L-shaped triominos look like this:

Input

The first line of input contains a single integer $n$ ( $1 \le n \le 10$ ), which is the $n$ of the description.

Each of the next $2^n$ lines contains $2^n$ integers $x$ ( $0 \le x \le \frac{4^n-1}{3}$ ), where 0 represents an empty space, and any positive number is a unique identifier of a triomino.

Output

Output a single integer, which is $1$ if the given grid is covered with $\frac{4^n-1}{3}$ unique triominos and a single empty space. Otherwise, output $0$ .

Examples2

Example 1
Input
```
2
1 1 2 2
1 3 3 2
4 4 3 5
4 0 5 5
```
Expected output
```
1
```
Example 2
Input
```
1
1 1
1 1
```
Expected output
```
0
```