
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 the sequence $\{a_n\}$ satisfies $a_1=1$, and $\frac{1}{1}a_1+ \frac{1}{2}a_2+ \frac{1}{3}a_3+…+ \frac{1}{n}a_n=a_{n+1}-1$ for all $n \in \mathbb{N}^*$, let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$.

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

$(2)$ Let $b_n= \frac{1}{S_n}$, and let $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the smallest positive integer $m$ such that $T_n < \frac{m}{10}$ holds for all $n \in \mathbb{N}^*$.

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