
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 an increasing sequence $\{a_n\}$, the sum of the first $n$ terms is denoted by $S_n$, and it is known that $S_{n-1} + S_n + S_{n+1} = 3n^2 + 2$ for $n \geqslant 2$ and $n \in \mathbb{N}^*$. Given $a_1 = 1$ and $S_{3k} = 324$, find the value of the positive integer $k$ that satisfies the given conditions.

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