
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 $A(-1,2,1)$, $B(1,2,1)$, and $C(-1,6,4)$,
(1) Find the area $S$ of the parallelogram with $\overrightarrow{AB}$ and $\overrightarrow{AC}$ as a pair of adjacent sides;
(2) If vector $\overrightarrow{a}$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and $|\overrightarrow{a}|=10$, find the coordinates of vector $\overrightarrow{a}$.

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