
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 $L_n$ be the $n$th Lucas number, where $L_1 = 1$, $L_2 = 3$ and for all integers $n \geq 2$, $L_{n + 1} = L_n + L_{n - 1}$. Compute the following product:
\[\prod_{k = 2}^{50} \left( \frac{L_k}{L_{k - 1}} - \frac{L_k}{L_{k + 1}} \right) = \frac{L_a}{L_b}\]for some positive integers $a$ and $b$.

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