
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
Let the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $T_{n}$, and $a_{1}=1$, $a_{n}=T_{n-1}$ for $n\geqslant 2$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ For any $n\in \mathbb{N}^{*}$, find the sum of the first $n$ terms of the sequence $\left\{\frac{n}{a_n}\right\}$.

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