
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 sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ is $S_n$, and it is given that $a_3=9$ and $S_6=60$.

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

$(2)$ If the sequence $\{b_n\}$ satisfies $b_{n+1}-b_{n}=a_{n}$ $(n\in\mathbb{N}_+)$ and $b_{1}=3$, find the sum of the first $n$ terms of the sequence $\left\{\frac{1}{{b_n}}\right\}$, denoted as $T_n$.

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