13. Let the sequence $\left\{a_{n}\right\}$ have the first term $a_{1}=1$, and the sum of the first $n$ terms $S_{n}$ and the general term $a_{n}$ satisfy $a_{n}=\frac{2 S_{n}^{2}}{2 S_{n}-1}$ $(n \geqslant 2)$, find: (1) the general formula for the sequence $\left\{S_{n}\right\}$; (2) the general formula for the sequence $\left\{a_{n}\right\}$.
13. (1) Since $a_{n}=S_{n}-S_{n-1}$, substituting $a_{n}=\frac{2 S_{n}^{2}}{2 S_{n}-1}$ and simplifying yields $S_{n-1}-2 S_{n} S_{n-1}-S_{n}=0$, hence $\frac{1}{S_{n}}-\frac{1}{S_{n-1}}=2$, so $\frac{1}{S_{n}}=\frac{1}{S_{1}}+2(n-1)$. When $n=1$, we get $S_{1}=a_{1}=1$, thus $\frac{1}{S_{n}}=2 n-1$, so $S_{n}=$ $\frac{1}{2 n-1}(n \geqslant 1)$
(2) Since $a_{n}=S_{n}-S_{n-1}=\frac{-2}{(2 n-1)(2 n-3)}(n \geqslant 2), a_{1}=1$, therefore
$$
a_{n}\left\{\begin{array}{l}
1, n=1 \\
\frac{-2}{(2 n-1)(2 n-3)}, n \geqslant 2 .
\end{array}\right.
$$