3.3.5 ** The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+a_{n}, n \in \mathbf{N}^{*}, b_{n}=\frac{1}{1+a_{n}}$,
$$
S_{n}=b_{1}+b_{2}+\cdots+b_{n}, P_{n}=b_{1} b_{2} \cdots b_{n},
$$

Find the value of $2 P_{n}+S_{n}$.