
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, $a_1=1$, and $3S_n=a_{n+1}-1$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $\{b_n\}$ be an arithmetic sequence whose sum of the first $n$ terms is $T_n$, $a_2=b_2$, $T_4=1+S_3$, find the minimum value of $\frac{2T_n+48}{n}$.

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