
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
In the arithmetic sequence $\{a_n\}$, it is given that $a_{15}+a_{16}+a_{17}=-45$ and $a_{9}=-36$. Let $S_n$ be the sum of the first $n$ terms.
$(1)$ Find the minimum value of $S_n$ and the corresponding value of $n$;
$(2)$ Calculate $T_n=|a_1|+|a_2|+\cdots+|a_n|$.

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