
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 $A(0,2,3)$, $B(-2,1,6)$, $C(1,-1,5)$.  
$(1)$ Find the area of the parallelogram with sides $AB$ and $AC$.  
$(2)$ If the vector $\overrightarrow{a}$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and $|\overrightarrow{a}|=\sqrt{3}$, find the coordinates of $\overrightarrow{a}$.

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