
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 $O$ be the origin, and $(1,1,1)$ is a fixed point. A plane passes through $(1,1,1)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at points $A$, $B$, and $C$, respectively, which are distinct from $O$. The coordinates of $A$, $B$, and $C$ are $(2,0,0)$, $(0,4,0)$, and $(0,0,6)$ respectively. Let $(p,q,r)$ be the center of the sphere passing through $A$, $B$, $C$, and $O$. Find
\[
\frac{1}{p} + \frac{1}{q} + \frac{1}{r}.
\]

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