
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 denoted as $S\_n$, it is known that $a\_1 = 2$ and the sequence formed by $4S\_2$, $3S\_3$, and $2S\_5$ is an arithmetic sequence.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. If the sequence $\{a\_n^2 \cdot b\_n\}$ is an arithmetic sequence with first term $1$ and common difference $2$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $T\_n$.

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