CF1794D.Counting Factorizations

The prime factorization of a positive integer $m$ is the unique way to write it as $\displaystyle m=p_1^{e_1}\cdot p_2^{e_2}\cdot \ldots \cdot p_k^{e_k}$ , where $p_1, p_2, \ldots, p_k$ are prime numbers, $p_1 < p_2 < \ldots < p_k$ and $e_1, e_2, \ldots, e_k$ are positive integers.

For each positive integer $m$ , $f(m)$ is defined as the multiset of all numbers in its prime factorization, that is $f(m)=\{p_1,e_1,p_2,e_2,\ldots,p_k,e_k\}$ .

For example, $f(24)=\{2,3,3,1\}$ , $f(5)=\{1,5\}$ and $f(1)=\{\}$ .

You are given a list consisting of $2n$ integers $a_1, a_2, \ldots, a_{2n}$ . Count how many positive integers $m$ satisfy that $f(m)=\{a_1, a_2, \ldots, a_{2n}\}$ . Since this value may be large, print it modulo $998\,244\,353$ .

The first line contains one integer $n$ ( $1\le n \le 2022$ ).

The second line contains $2n$ integers $a_1, a_2, \ldots, a_{2n}$ ( $1\le a_i\le 10^6$ ) — the given list.

Print one integer, the number of positive integers $m$ satisfying $f(m)=\{a_1, a_2, \ldots, a_{2n}\}$ modulo $998\,244\,353$ .

In the first sample, the two values of $m$ such that $f(m)=\{1,2,3,3\}$ are $m=24$ and $m=54$ . Their prime factorizations are $24=2^3\cdot 3^1$ and $54=2^1\cdot 3^3$ .

In the second sample, the five values of $m$ such that $f(m)=\{2,2,3,5\}$ are $200, 225, 288, 500$ and $972$ .

In the third sample, there is no value of $m$ such that $f(m)=\{1,4\}$ . Neither $1^4$ nor $4^1$ are prime factorizations because $1$ and $4$ are not primes.