
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 set of nonnegative integers to the set of nonnegative integers, such that:
\[ 3g(a^2 + b^2 + 1) = [g(a)]^3 + [g(b)]^3 \]
for all nonnegative integers \( a \) and \( b \).

Let \( n \) be the number of possible values of \( g(10) \), and let \( s \) be the sum of these possible values. Determine \( n \times s \).

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