This is the hard version of the problem. The only difference is that $$$1 \le x \le 5 \cdot 10^5$$$.

On the way home after buying his favorite soda "Zola Cero", Egor saw that elections for the position of "Best Number" are taking place in Saransk.

There are $$$n$$$ people at the polling station. Each person brought a number $$$a_i$$$. When the $$$i$$$-th person enters the voting booth, they choose a candidate that is a divisor of the number $$$a_i$$$. Let the chosen candidate be $$$p_i$$$.

After everyone has voted, we get an array of votes $$$[p_1, p_2, \ldots, p_n]$$$.

Egor really likes the number $$$x$$$ and considers the voting ideal if $$$x \cdot {lcm}(p_1, p_2, \ldots, p_n)$$$$$$^{\text{∗}}$$$ = $$$p_1 \cdot p_2 \cdot \ldots \cdot p_n$$$. Help him find the number of different$$$^{\text{†}}$$$ arrays $$$p$$$ modulo $$$10^9 + 7$$$ that are ideal.

$$$^{\text{∗}}$$$$$$lcm$$$ — least common multiple.

$$$^{\text{†}}$$$Two arrays of votes are considered different if there exists an index $$$i$$$ where the two arrays have different elements.

Input

The first line contains a single integer $$$t$$$ ($$$1 \leq t \leq 10^4$$$) — the number of test cases.

Then $$$t$$$ test cases follow.

The first line of each test case contains two integers $$$n$$$ and $$$x$$$ ($$$1 \leq n \leq 10^5$$$, $$$1 \leq x \leq 5 \cdot 10^5$$$) — the number of voters at the polling station and Egor's favorite number.

The second line of each test case contains $$$n$$$ integers: $$$a_1, a_2, \dots, a_n$$$ ($$$1 \leq a_i \leq 5 \cdot 10^5$$$) — the numbers brought by the voters.

It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$10^5$$$.

Output

For each test case, output the number of ways modulo $$$10^9 + 7$$$ to vote so that the resulting array of votes satisfies the condition.

Elections in Saransk (hard version)

Examples