
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 an arithmetic sequence $\{a_n\}$ with a common difference not equal to $0$, it satisfies that $a_1$, $a_2$, $a_7$ form a geometric sequence, and $a_3=9$.  
$(1)$ Find the general formula for $\{a_n\}$.  
$(2)$ If the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, find the sum of the first $n$ terms of the sequence $\left\{ \frac{S_n}{n} \right\}$, denoted as $T_n$.

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