
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
In triangle $ABC$, where $AB = 13$, $BC = 12$, and $CA = 5$. A circle $P$ with the smallest radius passes through $C$ and is tangent to $CA$ at its midpoint. Let points $Q$ and $R$ represent the intersection points, distinct from $C$, of circle $P$ with sides $AB$ and $BC$, respectively. The length of segment $QR$ is to be calculated.
A) $4$
B) $4.5$
C) $5$
D) $5.5$
E) $6$

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