Problem 1506

For any two positive integers $m$ and $n$, we define $l(m, n)$ as their least common multiple and $h(m, n)$ as their highest common factor. Given a prime $p > 3$, let $k$ denote the number of ordered pairs of positive integers $(m, n)$ satisfying the equation
$$l(m, n) + h(m, n) = p^4.$$
Determine the smallest possible value of $k$ across all choices of the prime $p > 3$.