
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
Solve the equation \( a^{2} + 2 = b! \) under the condition that \( a, b \) belong to the natural numbers. In the answer, specify the sum of the product of all possible \( a \) and the product of all possible \( b \) (if the equation has no solutions, indicate 0; if there are infinitely many solutions, indicate 1000).

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