
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 \(g\) be a function from the nonnegative integers to the nonnegative integers such that
\[2g(a^2 + b^2) = [g(a) + 1]^2 + [g(b) + 1]^2\]
for all nonnegative integers \(a\) and \(b.\) Determine the number of possible values of \(g(25)\), let \(n\) be this number, and let \(s\) be the sum of these possible values. Find \(n \times s.\)

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