12.6.5 Higher mathematics

Problem: A real number \( a \neq-1 \) is given. Let's define the sequence \( x_{n} \) as follows: \( x_{1}=a, x_{n+1}=x_{n}^{2}+x_{n}, n \geq 1 \). Let's also define the sequence \( y_{n} \), as well as the sum and product of its first \( n \) elements: \[ y_{n}=\frac{1}{1+x_{n}}, \quad S_{n}=\sum_{k=1}^{n} y_{k}, \quad p_{n}=\prod_{k=1}^{n} y_{k} . \]