
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, satisfying
\[2f(a^2 + b^2) = [f(a)]^2 + [f(b)]^2\]
for all nonnegative integers $a$ and $b$. Determine all possible values of $f(49)$, let $m$ be the number of such values, and let $t$ be their sum. Calculate $m \times t$.

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