
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 \(f\) be a function mapping nonnegative integers to nonnegative integers, such that
\[3f(a^2 + b^2) = [f(a)]^2 + 2[f(b)]^2\]
for all nonnegative integers \(a\) and \(b\).

Let \(n\) be the number of possible values for \(f(16)\), and let \(s\) be the sum of possible values for \(f(16)\). Find \(n \times s\).

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