
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

The sequence of integers \(\{a_i\}_{i=0}^{\infty}\) 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
\[
\sum_{n=0}^{\infty} \left( \frac{a_{n+3}}{a_{n+2}} - \frac{a_{n+2}}{a_n} + \frac{a_{n+1}}{a_{n+3}} - \frac{a_n}{a_{n+1}} \right).
\]

Present the answer in LaTex format: \boxed{Your answer}
user