
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 $\triangle PQR$, where $PQ=7$, $PR=9$, $QR=12$, and $S$ is the midpoint of $\overline{QR}$. What is the sum of the radii of the circles inscribed in $\triangle PQS$ and $\triangle PRS$?
A) $\frac{14\sqrt{5}}{13}$
B) $\frac{14\sqrt{5}}{6.5 + \sqrt{29}}$
C) $\frac{12\sqrt{4}}{8.5}$
D) $\frac{10\sqrt{3}}{7 + \sqrt{24}}$

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