
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 integers to the integers such that
\[f(m + n) + f(mn + 1) = f(m) f(n) + 2\]
for all integers $m$ and $n.$

Determine the number $n$ of possible values of $f(3)$ and the sum $s$ of all possible values of $f(3)$. Calculate $n \times s$.

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