HMMT 2019 A10

The sequence of integers $\left\{a_{i}\right\}_{i=0}^{\infty}$ satisfies $a_{0}=3, a_{1}=4$, and

$$a_{n+2}=a_{n+1} a_{n}+\left\lceil\sqrt{a_{n+1}^{2}-1} \sqrt{a_{n}^{2}-1}\right\rceil$$

for $n \geq 0$. Evaluate the sum

$$\sum_{n=0}^{\infty}\left(\frac{a_{n+3}}{a_{n+2}}-\frac{a_{n+2}}{a_{n}}+\frac{a_{n+1}}{a_{n+3}}-\frac{a_{n}}{a_{n+1}}\right) .$$