P16025 [ICPC 2021 NAC] You be The Judge, Again

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: :::align{center}  :::

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 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 $0$.