#P1624C. Division by Two and Permutation
Division by Two and Permutation
Description
You are given an array $a$ consisting of $n$ positive integers. You can perform operations on it.
In one operation you can replace any element of the array $a_i$ with $\lfloor \frac{a_i}{2} \rfloor$, that is, by an integer part of dividing $a_i$ by $2$ (rounding down).
See if you can apply the operation some number of times (possible $0$) to make the array $a$ become a permutation of numbers from $1$ to $n$ —that is, so that it contains all numbers from $1$ to $n$, each exactly once.
For example, if $a = [1, 8, 25, 2]$, $n = 4$, then the answer is yes. You could do the following:
- Replace $8$ with $\lfloor \frac{8}{2} \rfloor = 4$, then $a = [1, 4, 25, 2]$.
- Replace $25$ with $\lfloor \frac{25}{2} \rfloor = 12$, then $a = [1, 4, 12, 2]$.
- Replace $12$ with $\lfloor \frac{12}{2} \rfloor = 6$, then $a = [1, 4, 6, 2]$.
- Replace $6$ with $\lfloor \frac{6}{2} \rfloor = 3$, then $a = [1, 4, 3, 2]$.
The first line of input data contains an integer $t$ ($1 \le t \le 10^4$) —the number of test cases.
Each test case contains exactly two lines. The first one contains an integer $n$ ($1 \le n \le 50$), the second one contains integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$).
For each test case, output on a separate line:
- YES if you can make the array $a$ become a permutation of numbers from $1$ to $n$,
- NO otherwise.
You can output YES and NO in any case (for example, strings yEs, yes, Yes and YES will be recognized as a positive response).
Input
The first line of input data contains an integer $t$ ($1 \le t \le 10^4$) —the number of test cases.
Each test case contains exactly two lines. The first one contains an integer $n$ ($1 \le n \le 50$), the second one contains integers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^9$).
Output
For each test case, output on a separate line:
- YES if you can make the array $a$ become a permutation of numbers from $1$ to $n$,
- NO otherwise.
You can output YES and NO in any case (for example, strings yEs, yes, Yes and YES will be recognized as a positive response).
Samples
6
4
1 8 25 2
2
1 1
9
9 8 3 4 2 7 1 5 6
3
8 2 1
4
24 7 16 7
5
22 6 22 4 22
YES
NO
YES
NO
NO
YES
Note
The first test case is explained in the text of the problem statement.
In the second test case, it is not possible to get a permutation.