
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 $g$ that maps nonnegative integers to real numbers, with $g(0) = 0$ and $g(1) = 2$. The function satisfies the relation
\[ g(m + n) + g(m - n) = \frac{g(2m) + g(2n)}{2} \]
for all nonnegative integers $m \geq n$. Determine the sum of all possible values for $g(7)$.

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