
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 \( G(0) = 1 \), \( G(1) = 4 \), and
\[ G(n) = 3G(n-1) - 2G(n-2) \] for \( n \geq 2 \).
Find the sum
\[ \sum_{n=0}^\infty \frac{1}{G(2^n)} \].

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