
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 $\{a_n\}$, where $a_3=9$ and $a_8=29$.  
$(1)$ Find the general formula for the sequence $\{a_n\}$ and the expression for the sum of the first $n$ terms $S_n$.  
$(2)$ Let the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$ be $T_n$, find the value of $T_n$.

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