
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 from the nonnegative integers to nonnegative integers such that for all nonnegative integers \( a \) and \( b \):
\[ 3f(a^2 + b^2) = 2[f(a)]^2 + 2[f(b)]^2 - f(a)f(b) \]
Determine the number of possible values \( n \) and their sum \( s \) for \( f(16) \), and calculate \( n \times s \).

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