Sets, Relations and Functions: JEE Main 2024 Mathematics question

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