
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 let $(a, b, c)$ be the midpoint of the line segment connecting $O$ and a point $(d, e, f)$. A plane passes through $(d, e, f)$ and intersects the $x$-axis, $y$-axis, and $z$-axis at points $A$, $B$, and $C$ respectively, all distinct from $O$. Let $(p, q, r)$ be the center of the sphere passing through $A$, $B$, $C$, and $O$. Find
\[
\frac{d}{p} + \frac{e}{q} + \frac{f}{r}.
\]

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