
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
Circle $P$ with radius 3 has diameter $\overline{XY}$. Circle $Q$ is internally tangent to circle $P$ at point $X$. Circle $R$ is externally tangent to circle $P$, externally tangent to circle $Q$, and tangent to $\overline{XY}$. The radius of circle $Q$ is twice the radius of circle $R$, and can be written in the form $\sqrt{p}-q$, where $p$ and $q$ are positive integers. Find $p+q$.

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