
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 $g$ for nonnegative integers such that $g(2) = 4$, and for all nonnegative integers $m \geq n$, the equation holds:
\[ g(m + n) + g(m - n) = \frac{g(3m) + g(3n)}{3} \]
Find the sum of all possible values of $g(12)$.

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