
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
Call an ordered pair \((a, b)\) of positive integers fantastic if and only if \(a, b \leq 10^{4}\) and
\[
\operatorname{gcd}(a \cdot n! - 1, a \cdot (n+1)! + b) > 1
\]
for infinitely many positive integers \(n\). Find the sum of \(a+b\) across all fantastic pairs \((a, b)\).

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