Mathematics Question from JEE Main 2024 · 30 January, Shift 1

Let $A = \{1, 2, 3, \ldots, 7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f : A \to P(A)$ such that $a \in f(a)$ for all $a \in A$ is $m^n$, where $m, n \in \mathbb{N}$ and $m$ is least, then $m + n$ is equal to _____.