
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 a fixed point. A plane passes through \((a,b,c)\) and intersects the \(x\)-axis at \((d,0,0)\), the \(y\)-axis at \((0,e,0)\), and the \(z\)-axis at \((0,0,f)\), where \(d\), \(e\), and \(f\) are non-zero constants, all distinct from \(O\). Let \((p,q,r)\) be the center of the sphere passing through \(O\), \(A\), \(B\), and \(C\). Find
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]

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