
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 for all nonnegative integers \( a \), \( b \), and \( c \):
\[ 3g(a^2 + b^2 + c^2) = [g(a)]^2 + [g(b)]^2 + [g(c)]^2. \]
Find the possible values of \( g(27) \), denote the number of these values as \( m \), and their sum as \( t \). Calculate \( m \times t \).

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