10. For any positive integers $m, n$, define the function $f(m, n)$ as follows:
(1) $f(1,1)=1$;
(2) $f(m+1, n)=f(m, n)+2(m+n)$;
(3) $f(m, n+1)=f(m, n)+2(m+n-1)$.
(1) Find the explicit formula for $f(m, n)$;
(2) Let $a_{n}=\frac{\sqrt{f(n, n)}}{2^{n-1}}\left(n \in \mathbf{N}^{*}\right), S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$, prove: $S_{n}<6$.