
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 a sequence defined by $G_0 = 1$, $G_1 = 2$, and $G_{n+2} = G_{n+1} + 2G_n$. Compute the sum:
\[
\sum_{n=0}^\infty \frac{G_n}{5^n}.
\]

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