
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 $(2a, 2b, 2c)$ be a fixed point. A plane passes through $(2a, 2b, 2c)$ and intersects the x-axis, y-axis, and z-axis at points $A$, $B$, and $C$, respectively, all distinct from $O$. Simultaneously, $(2a, 2b, 2c)$ lies on a cylinder whose central axis is along the line joining $O$ to $(p,q,r)$, and whose radius equals the distance from $O$ to $(2a, 2b, 2c)$. Find:

\[ \frac{2a}{p} + \frac{2b}{q} + \frac{2c}{r} \]

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