
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 4 has diameter $\overline{XY}$. Circle $Q$ is internally tangent to circle $P$ at $X$. Circle $R$ is internally tangent to circle $P$, externally tangent to circle $Q$, and tangent to $\overline{XY}$. The radius of circle $Q$ is four times the radius of circle $R$. Find the radius of circle $Q$ in the simplest form, which can be written as $\sqrt{a}-b$, where $a$ and $b$ are positive integers. Provide the value of $a+b$.

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