
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 three points in space $A(2,0,-2)$, $B(1,-1,-2)$, and $C(3,0,-4)$, let $\overrightarrow{a}=\overrightarrow{AB}$ and $\overrightarrow{b}=\overrightarrow{AC}$.  
$(1)$ If $|\overrightarrow{c}|=3$ and $\overrightarrow{c}$ is parallel to $\overrightarrow{BC}$, find the vector $\overrightarrow{c}$.  
$(2)$ Find the area $S$ of the parallelogram with $\overrightarrow{a}$ and $\overrightarrow{b}$ as adjacent sides.

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