
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 $a_2+a_3+a_4=-3$ and $a_1=-5$:
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ Define the sum of the first $n$ terms of the sequence $\{|a_n|\}$ as $T_n$, and find $T_{50}$;
$(3)$ Determine whether there exists a positive integer $k$, such that $\frac{a_ka_{k+1}}{a_{k+2}}$ is also a term in the sequence $\{a_n\}$. If such $k$ exists, find all values of $k$; if not, explain why.

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