
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
For an arithmetic sequence $\{a_n\}$ with a common difference not equal to $0$, the sum of the first $n$ terms is $S_n$. It is known that $S_4=10$, and $a_1$, $a_3$, $a_9$ form a geometric sequence.  
$(1)$ Find the general formula for $\{a_n\}$.  
$(2)$ Find the sum of the first $n$ terms $T_n$ for the sequence $\left\{ \frac{a_n}{3^n} \right\}$.

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