
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 point $A$, $y$-axis at point $B$, and $z$-axis at point $C$, each different from $O.$ Let $(p,q,r)$ be the center of the sphere passing each point $O$, $A$, $B$, and $C$. Suppose the sphere’s center coordinates relate to the axis intercepts as $p^3 = \alpha$, $q^3 = \beta$, and $r^3 = \gamma$. Find:
\[
\frac{a}{p^3} + \frac{b}{q^3} + \frac{c}{r^3}.
\]

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