It Telescopes?

Calculus Level 3

Let $\{a_{n}\}$ be a sequence of real numbers satisfying $\begin{cases} a_{0}=1 \\ a_{n+1}=\sqrt{4+3a_{n}+a_{n}^{2}}-2 & \text{for } n \ge 0. \end{cases}$ Let $\displaystyle S=\sum_{n=0}^{\infty} a_{n}$ .

If $S$ converges, submit your answer as $\big\lfloor 100S \big\rfloor$ .
If $S$ diverges, submit your answer as $-1$ .

This problem is based on a recent Putnam contest problem.