
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 the function $f$ mapping nonnegative integers to real numbers with $f(1) = 2$, and the equation
\[ f(m+n) + f(m-n) = 3f(m)f(n) \]
holds for all nonnegative integers $m \geq n$.
Find $f(10)$.

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