A13158. Beautiful Numbers

You are given a permutation $p=[p_1, p_2, \ldots, p_n]$ of integers from $1$ to $n$ . Let's call the number $m$ ( $1 \le m \le n$ ) beautiful, if there exists two indices $l, r$ ( $1 \le l \le r \le n$ ), such that the numbers $[p_l, p_{l+1}, \ldots, p_r]$ is a permutation of numbers $1, 2, \ldots, m$ .

For example, let $p = [4, 5, 1, 3, 2, 6]$ . In this case, the numbers $1, 3, 5, 6$ are beautiful and $2, 4$ are not. It is because:

- if $l = 3$ and $r = 3$ we will have a permutation $[1]$ for $m = 1$ ;
- if $l = 3$ and $r = 5$ we will have a permutation $[1, 3, 2]$ for $m = 3$ ;
- if $l = 1$ and $r = 5$ we will have a permutation $[4, 5, 1, 3, 2]$ for $m = 5$ ;
- if $l = 1$ and $r = 6$ we will have a permutation $[4, 5, 1, 3, 2, 6]$ for $m = 6$ ;
- it is impossible to take some $l$ and $r$ , such that $[p_l, p_{l+1}, \ldots, p_r]$ is a permutation of numbers $1, 2, \ldots, m$ for $m = 2$ and for $m = 4$ .

You are given a permutation $p=[p_1, p_2, \ldots, p_n]$ . For all $m$ ( $1 \le m \le n$ ) determine if it is a beautiful number or not.

The first line contains the only integer $t$ ( $1 \le t \le 1000$ ) — the number of test cases in the input. The next lines contain the description of test cases.

The first line of a test case contains a number $n$ ( $1 \le n \le 2 \cdot 10^5$ ) — the length of the given permutation $p$ . The next line contains $n$ integers $p_1, p_2, \ldots, p_n$ ( $1 \le p_i \le n$ , all $p_i$ are different) — the given permutation $p$ .

It is guaranteed, that the sum of $n$ from all test cases in the input doesn't exceed $2 \cdot 10^5$ .

Print $t$ lines — the answers to test cases in the order they are given in the input.

The answer to a test case is the string of length $n$ , there the $i$ -th character is equal to $1$ if $i$ is a beautiful number and is equal to $0$ if $i$ is not a beautiful number.

The first test case is described in the problem statement.

In the second test case all numbers from $1$ to $5$ are beautiful:

- if $l = 3$ and $r = 3$ we will have a permutation $[1]$ for $m = 1$ ;
- if $l = 3$ and $r = 4$ we will have a permutation $[1, 2]$ for $m = 2$ ;
- if $l = 2$ and $r = 4$ we will have a permutation $[3, 1, 2]$ for $m = 3$ ;
- if $l = 2$ and $r = 5$ we will have a permutation $[3, 1, 2, 4]$ for $m = 4$ ;
- if $l = 1$ and $r = 5$ we will have a permutation $[5, 3, 1, 2, 4]$ for $m = 5$ .