
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
Let $O = (1,1,1)$ be a point. A variable plane has a distance of 2 from the point $O$, and intersects the $x$-axis, $y$-axis, and $z$-axis at $A$, $B$, and $C$, respectively, all distinct from $O$. Let $(p,q,r)$ be the centroid of triangle $ABC$. Find
\[\frac{1}{p^2} + \frac{1}{q^2} + \frac{1}{r^2}.\]

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