
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 $\{a_n\}$ be a geometric sequence, and let $\{b_n\}$ be a sequence satisfying $b_n=na_1+(n-1)a_2+\ldots+2a_{n-1}+a_n$, where $n\in\mathbb{N}^*$, and it's given that $b_1=m$, $b_2=\frac{3m}{2}$, where $m\neq 0$.
(1) Find the general term of the sequence $\{a_n\}$ (expressed in terms of $m$);
(2) 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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