
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
Let $F_n$ be the $n$th Fibonacci number, where $F_3 = 2$, $F_4 = 3$, and $F_{n + 1} = F_n + F_{n - 1}$ for $n \geq 4.$ Calculate
\[
\prod_{k = 4}^{150} \left( \frac{F_k^2 + F_{k+1}}{F_{k-1} F_{k+1}} - \frac{F_k F_{k+2}}{F_{k+1}^2} \right).
\]
Find this product in terms of Fibonacci numbers, $\frac{F_a}{F_b}$, for some integers $a$ and $b.$ Determine the ordered pair $(a, b)$.

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