10. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{n+1}=a_{n}+a_{n}^{2}\left(n \in \mathbf{N}^{*}\right)$. Let
$$
S_{n}=\frac{1}{\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)}, \quad T_{n}=\sum_{k=1}^{n} \frac{1}{1+a_{k}}
$$

Find the value of $S_{n}+T_{n}$.