
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(0,1,5), B(1,5,0), and C(5,0,1), if the vector $\vec{a}=(x,y,z)$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and the magnitude of vector $\vec{a}$ is $\sqrt{15}$, then find the value of $x^2y^2z^2$.

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