Let $m$ and $n$ be positive integers. If $m$ has exactly $7$ positive divisors, $n$ has exactly $10$ positive divisors, and $mn$ has exactly $22$ positive divisors, then how many divisors does $m^2 n^2$ have?

Let $m$ and $n$ be positive integers. If $m$ has exactly $7$ positive divisors, $n$ has exactly $10$ positive divisors, and $mn$ has exactly $22$ positive divisors, then how many divisors does $m^2 n^2$ have?