
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
Consider a geometric sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. Given that $a\_1 + a\_3 = 30$, and the sequence $3S\_1, 2S\_2, S\_3$ forms an arithmetic sequence.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. If a sequence $\{b\_n\}$ satisfies $b\_1 = 3$ and $b\_{n+1} - 3b\_n = 3a\_n$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $B\_n$.
3. Remove the 3rd, 6th, 9th, ..., $3n$-th terms from the sequence $\{a\_n\}$, and the remaining terms form a new sequence $\{c\_n\}$, with the sum of its first $n$ terms denoted as $T\_n$. If $\frac{T\_{n+1}}{T\_n} > a$ holds for any $n \in \mathbb{N}^+$, find the maximum value of the real number $a$.

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