
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 $C$ be a circle with radius $2$. Two points are chosen independently at random on the circumference of $C$. The probability that the straight-line (chord) distance between these points is at least $3$ is $\dfrac{p - q\pi}{r}$, where $p$, $q$, and $r$ are positive integers with $\gcd(p, q, r) = 1$. What is $p + q + r$?

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