
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 a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. The first term $a\_1 = 1$, and $3S\_n = a_{n+1} - 1$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $\{b\_n\}$ be an arithmetic sequence with the sum of its first $n$ terms denoted as $T\_n$. Given that $a\_2 = b\_2$ and $T\_4 = 1 + S\_3$, find the minimum value of $\frac{2T\_n + 48}{n}$.

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