
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
Define $g$ as a function from the integers to the integers such that 
\[ g(m + n) + g(mn - 1) = g(m)g(n) - g(m) + g(n) + 1 \]
for all integers $m$ and $n$. Assume $g(0) = 0$. Determine the quantity and sum of all possible values of $g(2)$, and hence find the product of these two results.

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