
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 sum of the first $n$ terms of the arithmetic sequence $\{a\_n\}$ is $S\_n$, and it satisfies $S\_5 = 30$ and $S\_{10} = 110$. The sum of the first $n$ terms of the sequence $\{b\_n\}$ is $T\_n$, and it satisfies $T\_n = \frac{3}{2}b\_n - \frac{1}{2}$ for all $n \in \mathbb{N}^*$.

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

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