
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
The sequence $\{a_{n}\}$ satisfies $a_{1}+\frac{1}{2}{a_2}+\frac{1}{3}{a_3}+…+\frac{1}{n}{a_n}={a_{n+1}}-1$, $n\in N^{*}$, and $a_{1}=1$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $S_{n}=a_{1}\cdot a_{n}+a_{2}\cdot a_{n-1}+a_{3}\cdot a_{n-2}+\ldots +a_{n}\cdot a_{1}$, $b_{n}=\frac{1}{{3{S_n}}}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$, find the smallest positive integer $m$ such that $T_{n} \lt \frac{m}{{2024}}$ holds for any $n\in N*$.<br/>$(Reference formula: $1^{2}+2^{2}+3^{2}+\ldots +n^{2}=\frac{{n({n+1})({2n+1})}}{6}$, $n\in N*)$

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