Description:
You are given $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$. Each of $$$a_i$$$ has between $$$3$$$ and $$$5$$$ divisors. Consider $$$a = \prod a_i$$$ — the product of all input integers. Find the number of divisors of $$$a$$$. As this number may be very large, print it modulo prime number $$$998244353$$$.
Input Format:
The first line contains a single integer $$$n$$$ ($$$1 \leq n \leq 500$$$) — the number of numbers.
Each of the next $$$n$$$ lines contains an integer $$$a_i$$$ ($$$1 \leq a_i \leq 2\cdot 10^{18}$$$). It is guaranteed that the number of divisors of each $$$a_i$$$ is between $$$3$$$ and $$$5$$$.
Output Format:
Print a single integer $$$d$$$ — the number of divisors of the product $$$a_1 \cdot a_2 \cdot \dots \cdot a_n$$$ modulo $$$998244353$$$.
Hacks input
For hacks, the input needs to be provided in a special format.
The first line contains an integer $$$n$$$ ($$$1 \leq n \leq 500$$$) — the number of numbers.
Each of the next $$$n$$$ lines contains a prime factorization of $$$a_i$$$. The line contains an integer $$$k_i$$$ ($$$2 \leq k_i \leq 4$$$) — the number of prime factors of $$$a_i$$$ and $$$k_i$$$ integers $$$p_{i,j}$$$ ($$$2 \leq p_{i,j} \leq 2 \cdot 10^{18}$$$) where $$$p_{i,j}$$$ is the $$$j$$$-th prime factor of $$$a_i$$$.
Before supplying the input to the contestant, $$$a_i = \prod p_{i,j}$$$ are calculated. Note that each $$$p_{i,j}$$$ must be prime, each computed $$$a_i$$$ must satisfy $$$a_i \leq 2\cdot10^{18}$$$ and must have between $$$3$$$ and $$$5$$$ divisors. The contestant will be given only $$$a_i$$$, and not its prime factorization.
For example, you need to use this test to get the first sample:
Note:
In the first case, $$$a = 19305$$$. Its divisors are $$$1, 3, 5, 9, 11, 13, 15, 27, 33, 39, 45, 55, 65, 99, 117, 135, 143, 165, 195, 297, 351, 429, 495, 585, 715, 1287, 1485, 1755, 2145, 3861, 6435, 19305$$$ — a total of $$$32$$$.
In the second case, $$$a$$$ has four divisors: $$$1$$$, $$$86028121$$$, $$$86028157$$$, and $$$7400840699802997 $$$.
In the third case $$$a = 202600445671925364698739061629083877981962069703140268516570564888699 375209477214045102253766023072401557491054453690213483547$$$.
In the fourth case, $$$a=512=2^9$$$, so answer equals to $$$10$$$.