
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 $\overrightarrow{BA}=(-1,-2,2)$, $\overrightarrow{BC}=(-2,2,1)$, $\overrightarrow{AC}=(-1,4,-1)$, let $\overrightarrow{a}=\overrightarrow{BA}$, $\overrightarrow{b}=\overrightarrow{BC}$, $\overrightarrow{c}=\overrightarrow{AC}$.  
$(1)$ Determine the shape of $\triangle ABC$;  
$(2)$ If $(-2\overrightarrow{a}+k\overrightarrow{b})∥\overrightarrow{c}$, find the value of $k$.

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