
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 being $S_n$, the first term being $a_1$, and $1$, $a_n$, $S_n$ forming an arithmetic sequence.

    (1) Find the general formula for the sequence $\{a_n\}$:

    (2) Let $T_n$ be the sum of the first $n$ terms of the sequence $\left\{\frac{n}{a_n}\right\}$. If for all $n\in \mathbb{N}^*$, it always holds that $T_n < \frac{m-4}{3}$, where $m\in \mathbb{N}^*$, find the minimum value of $m$.

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