Problem:
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.

Sample Input 1

2 6

Sample Output 1

4

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

Sample Input 2

3 12

Sample Output 2

18

Sample Input 3

100000 1000000000

Sample Output 3

957870001

Solution in Haskell (GHC 7.10.3) (judge verdict: Runtime Error)

import qualified Data.IntMap as M

main :: IO ()
main = do
    [n, m] <- map read . words <$> getLine
    print $ solve n m

solve :: Int -> Int -> Int
solve n = foldr (%*%) 1 . map (\b -> comb (b + n - 1) b) . M.elems . factorization
    where x %*% y = (x * y) `mod` 1000000007

factorization :: Int -> M.IntMap Int
factorization 1 = M.empty
factorization n = M.insertWith (+) p 1 $ factorization (n `div` p)
    where p = head $ filter (\k -> n `mod` k == 0) [2 ..]

comb :: Int -> Int -> Int
comb n k
    | k < 0 = 0
    | k > n = 0
    | otherwise = fact n `div` (fact k * fact (n - k))

fact :: Int -> Int
fact 0 = 1
fact n = n * fact (n - 1)