
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 as usual with $F_1 = F_2 = 1$ and $F_{n + 1} = F_n + F_{n - 1}.$ Consider the product 
\[\prod_{k = 3}^{50} \left( \frac{F_k}{F_{k - 2}} - \frac{F_k}{F_{k + 2}} \right).\]
Express this product in the form $\frac{F_c}{F_d}$ for some integers $c$ and $d.$ What is the ordered pair $(c,d)?$

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