
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
Given the sequence $\{a_n\}$, where $a_1 = -2$, and the sum of the first $n$ terms, $S_n$, satisfies the equation $a_{n+1} + 3S_n + 2 = 0$ for all $n \in \mathbb{N}^*$.

1. Find the general term formula for the sequence $\{a_n\}$.
2. Determine if there exists a pair of integers $(m, n)$ such that $a_n^2 - ma_n - 4m - 8 = 0$. If such pairs exist, find all pairs that satisfy the condition; otherwise, explain why no such pairs exist.

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