Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition $(S, T)$ of $X$ $f$-balanced if for all $s \in S$, we have $f(s) \in T$, and for all $t \in T$, we have $f(t) \in S$. 

A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition.

Let $g(f)$ be the number of $f$-balanced partitions, and let $m$ equal the maximum value of $g$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.

Let's think step by step and output the final answer within \boxed{}.
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