
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 $n$th term of a sequence defined by $G_1 = 1, G_2 = 3$, and $G_{n+1} = G_n + 2G_{n-1}$ for all $n \geq 2$. Compute the value of:
\[\prod_{k = 3}^{50} \left( \frac{G_k}{G_{k - 1}} - \frac{G_k}{G_{k + 1}} \right).\]

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