Let $A_{n}=\{1,2, \ldots, n\}$, where $n \geq 3$. Let $\mathcal{F}$ be the family of all non-constant functions $f: A_{n} \rightarrow A_{n}$ satisfying the following conditions:\n(1) $f(k) \leq f(k+1)$ for $k=1,2, \ldots, n-1$,\n(2) $f(k)=f(f(k+1))$ for $k=1,2, \ldots, n-1$.\nFind the number of functions in $\mathcal{F}$. for $n=3$

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