13. (13 points) In the sequence $\left\{a_{n}\right\}$,
$$
a_{1}=1, a_{2}=\frac{1}{4} \text {, }
$$

and $a_{n+1}=\frac{(n-1) a_{n}}{n-a_{n}}(n=2,3, \cdots)$.
(1) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(2) Prove that for all $n \in \mathbf{Z}_{+}$, we have
$$
\sum_{k=1}^{n} a_{k}^{2}<\frac{7}{6} .
$$