
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 double of Fibonacci sequence values, that is, $G_n = 2 \cdot F_n$ where $F_n$ are Fibonacci numbers with $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$. Compute
\[
\sum_{n=0}^{\infty} \frac{G_n}{5^n}.
\]

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