Divine Divisor

An integer $N > 1$ is given. For an integer $d > 1$, we say that $d$ is a divisor of $N$ with multiplicity $k > 0$ (where $k$ is an integer) if $d^k \mid N$ but $d^{k+1} \nmid N$. For example, $N = 48 = 2^4 \cdot 3$ has, among others, the divisor $2$ with multiplicity $4$, the divisor $3$ with multiplicity $1$, the divisor $4$ with multiplicity $2$, and the divisor $6$ with multiplicity $1$.

We call $d$ a divine divisor of $N$ if $d$ is a divisor of $N$ with some multiplicity $k$ and $N$ has no divisor whose multiplicity is greater than $k$. In other words, a divine divisor is a divisor that attains the maximum multiplicity over all divisors of $N$. For example, the only divine divisor of $48$ is $2$ (with multiplicity $4$), while the divine divisors of $6$ are $2$, $3$, and $6$ (each with multiplicity $1$).

Determine the maximum multiplicity $k$ over all divisors of $N$, and the number of divine divisors of $N$.

$N$ is given in a slightly unusual form. The first line contains a single integer $n$ ($1 \le n \le 600$). The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($2 \le a_i \le 10^{18}$), separated by single spaces. They define $N = a_1 \cdot a_2 \cdot \cdots \cdot a_n$.

Print two lines. The first line contains the largest integer $k$ for which some divisor $d > 1$ of $N$ satisfies $d^k \mid N$. The second line contains the number of divine divisors of $N$, that is, the number of divisors whose multiplicity equals $k$. This value can be very large, so print it as an exact integer.