
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 $G_n$ be the nth Fibonacci number with the usual conditions $G_1 = G_2 = 1$ and $G_{n+1} = G_n + G_{n-1}$. Calculate the following and express it as a fraction of two Fibonacci numbers:
\[
\prod_{k=3}^{50} \left( \frac{G_k + G_{k+2}}{G_{k-1}} - \frac{G_k + G_{k-2}}{G_{k+1}} \right)
\]

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