
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 not on the origin. A plane passes through $(a, b, c)$ and intersects the $x$-axis at $(2a, 0, 0)$, the $y$-axis at $(0, 2b, 0)$, and the $z$-axis at $(0, 0, 2c)$, all distinct from $O$. Let $(p, q, r)$ be the center of the sphere passing through $O$, $(2a, 0, 0)$, $(0, 2b, 0)$, and $(0, 0, 2c)$. Determine the value of
\[
\frac{a}{p} + \frac{b}{q} + \frac{c}{r}.
\]

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