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

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