
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
Define a sequence where $G(0) = 0,$ $G(1) = \frac{5}{4},$ and
\[G(n) = 3G(n - 1) - \frac{1}{2} G(n - 2)\] for $n \ge 2.$ Find
\[\sum_{n = 0}^\infty \frac{1}{G(3^n)}.\]

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