
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 that the sequence $\{a\_n\}$ is a geometric sequence, and the sequence $\{b\_n\}$ satisfies $b\_n = n a\_1 + (n-1) a\_2 + \ldots + 2 a\_{n-1} + a\_n$, where $n \in \mathbb{N}^*$, and it is known that $b\_1 = m$, $b\_2 = \frac{3m}{2}$, where $m \neq 0$.

1. Find the first term and common ratio of the sequence $\{a\_n\}$;
2. When $m = 1$, find $b\_n$;
3. Let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n\}$. If for any positive integer $n$, $S\_n \in [1, 3]$, find the range of values for the real number $m$.

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