
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
\[ 3g(a^2 + b^2) = g(a)g(b) + 2[g(a) + g(b)] \]
for all nonnegative integers \( a \) and \( b \).

Let \( m \) be the number of possible values of \( g(50) \), and let \( t \) be the sum of the possible values of \( g(50) \). Find \( m \times t \).

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