
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, $y$-axis, and $z$-axis at points $A$, $B$, and $C$, respectively, which are distinct from $O$. If $(p, q, r)$ represents the center of the sphere passing through points $A$, $B$, $C$, and $O$, and the sphere has a radius $R$, find the value of
\[\frac{a^2}{p^2} + \frac{b^2}{q^2} + \frac{c^2}{r^2}.\]

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