D - Factorization

Score : $400$ points

Problem Statement

You are given positive integers $N$ and $M$.

How many sequences $a$ of length $N$ consisting of positive integers satisfy $a_1 \times a_2 \times ... \times a_N = M$? Find the count modulo $10^9+7$.

Here, two sequences $a'$ and $a''$ are considered different when there exists some $i$ such that $a_i' \neq a_i''$.

Constraints

• All values in input are integers.
• $1 \leq N \leq 10^5$
• $1 \leq M \leq 10^9$

Input

Input is given from Standard Input in the following format:

$N$ $M$

Output

Print the number of the sequences consisting of positive integers that satisfy the condition, modulo $10^9 + 7$.

2 6

4

Four sequences satisfy the condition: $\{a_1, a_2\} = \{1, 6\}, \{2, 3\}, \{3, 2\}$ and $\{6, 1\}$.

3 12

18

100000 1000000000

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