SEEMOUS 2014 Problem 2

Consider the sequence $(x_n)$ given by $$x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2.$$ Prove that the sequence $y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1$ is convergent and find its limit.