A14375. Moamen and XOR

Moamen and Ezzat are playing a game. They create an array $a$ of $n$ non-negative integers where every element is less than $2^k$ .

Moamen wins if $a_1 \,\&\, a_2 \,\&\, a_3 \,\&\, \ldots \,\&\, a_n \ge a_1 \oplus a_2 \oplus a_3 \oplus \ldots \oplus a_n$ .

Here $\&$ denotes the [bitwise AND operation](https://en.wikipedia.org/wiki/Bitwise_operation#AND), and $\oplus$ denotes the [bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR).

Please calculate the number of winning for Moamen arrays $a$ .

As the result may be very large, print the value modulo $1\,000\,000\,007$ ( $10^9 + 7$ ).

The first line contains a single integer $t$ ( $1 \le t \le 5$ )— the number of test cases.

Each test case consists of one line containing two integers $n$ and $k$ ( $1 \le n\le 2\cdot 10^5$ , $0 \le k \le 2\cdot 10^5$ ).

For each test case, print a single value — the number of different arrays that Moamen wins with.

Print the result modulo $1\,000\,000\,007$ ( $10^9 + 7$ ).

In the first example, $n = 3$ , $k = 1$ . As a result, all the possible arrays are $[0,0,0]$ , $[0,0,1]$ , $[0,1,0]$ , $[1,0,0]$ , $[1,1,0]$ , $[0,1,1]$ , $[1,0,1]$ , and $[1,1,1]$ .

Moamen wins in only $5$ of them: $[0,0,0]$ , $[1,1,0]$ , $[0,1,1]$ , $[1,0,1]$ , and $[1,1,1]$ .