
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 points \( A(0,2,3) \), \( B(-2,1,6) \), and \( C(1,-1,5) \):

(1) Find the area of the parallelogram with sides \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \).

(2) If \( |\vec{a}| = \sqrt{3} \), and \( \vec{a} \) is perpendicular to both \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \), find the coordinates of vector \( \vec{a} \).

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