
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, defined by \( F_1 = F_2 = 1 \) and \( F_{n + 1} = F_n + F_{n - 1} \). Evaluate the product
\[
\prod_{k = 3}^{50} \left( \frac{F_k}{F_{k - 1}} - \frac{F_k}{F_{k + 1}} \right)
\]
and express the result as \( \frac{F_a}{F_b} \) for some positive integers \( a \) and \( b \). Find the ordered pair \( (a, b) \).

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