
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 a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, and it satisfies $a_1=-2$, $a_{n+1}+3S_n+2=0$ ($n\in\mathbb{N}^*$).

1. Find the values of $a_2$ and $a_3$,
2. Find the general formula for the terms of the sequence $\{a_n\}$,
3. Determine if there are any integer pairs (m, n) such that the equation $a_n^2 - m \cdot a_n = 4m + 8$ is satisfied. If they exist, find all such (m, n) pairs; if not, explain why.

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