1557C · Moamen and XOR

Problem: 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, and $\oplus$ denotes the bitwise XOR operation.

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

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

Input Format: 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 $k$ ($1 \le n\le 2\cdot 10^5$, $0 \le k \le 2\cdot 10^5$).

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

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

Note: 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 $[1,1,1]$.