
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 \( (a, b, c) \) be a fixed point. A plane passes through \( (a,b,c) \) and intersects the \( x \)-axis, \( y \)-axis, and \( z \)-axis at \( A, B, \) and \( C \), respectively, all distinct from \( O \). Let \( (d, e, f) \) be another distinct point not lying on the axes or the origin. Let \( (p, q, r) \) be the center of the sphere passing through \( A, B, C, O, \) and \( (d, e, f) \). Find
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]

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