
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
Define a modified Fibonacci sequence $\{F_n\}$ where $F_1 = 2$, $F_2 = 1$, and for all integers $n \geq 3$, $F_{n + 1} = F_n + F_{n - 1}$. Determine the value of the expression:
\[\prod_{k = 3}^{101} \left( \frac{F_k}{F_{k - 1}} - \frac{F_k}{F_{k + 1}} \right)\]
and express it as a ratio $\frac{F_a}{F_b}$ for some positive integers $a$ and $b$. Enter the ordered pair $(a, b)$.

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