
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 \( F_1 = F_2 = 1 \) and \( F_{n+1} = F_n + F_{n-1} \). Calculate:
\[
\prod_{k=3}^{101} \left( \frac{F_k}{F_{k-1}} - \frac{F_k}{F_{k+1}} \right)
\]
and express it as \(\frac{F_a}{F_b}\) for some positive integers \(a\) and \(b\). What is the ordered pair \((a, b)\)?

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