
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
Let the sequence $\{a_n\}$ be a geometric sequence, and the sequence $\{b_n\}$ satisfies $b_n = na_1 + (n-1)a_2 + \ldots + 2a_{n-1} + a_n$, where $n \in \mathbb{N}^*$, and it is known that $b_1 = m$, $b_{2} = \frac{3m}{2}$, where $m \neq 0$.

(Ⅰ) Find the first term and the common ratio of the sequence $\{a_n\}$.

(Ⅱ) When $m=1$, find $b_n$.

(Ⅲ) Let $S_n$ be 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 the real number $m$.

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