
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 first term $a_1 = a$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$, satisfying: $$S_n^2 = 3n^2a_n + S_{n-1}^2$$, $a_n \neq 0$, $n \geq 2$, $n \in \mathbb{N}^*$.  
(1) If the sequence $\{a_n\}$ is an arithmetic sequence, find the value of $a$;  
(2) Determine the set $M$ of values for $a$ such that when $a \in M$, the sequence $\{a_n\}$ is increasing.

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