
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 arithmetic sequence $\{a\_n\}$ with the sum of its first $n$ terms, $S\_n$, satisfying $S\_5=30$ and $S\_{10}=110$. Another sequence $\{b\_n\}$ has the sum of its first $n$ terms, $T\_n$, satisfying $T\_n=\frac{3}{2}b\_n-\frac{1}{2} (n\in\mathbb{N}^*)$.

1. Find $S\_n$ and $b\_n$.
2. Compare $S\_n b\_n$ and $T\_n a\_n$, and explain the reasoning.

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