
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 $ABC$, $AB = AC = 144$, and $BC = 80$. Circle $P$ has radius $24$ and is tangent to $\overline{AC}$ and $\overline{BC}$. A second circle $Q$ is externally tangent to $P$ and is tangent to both $\overline{AB}$ and $\overline{BC}$. Assume no point of circle $Q$ lies outside of $\triangle ABC$. Express the radius of circle $Q$ as $m - n \sqrt{k}$, where $m$, $n$, and $k$ are integers and $k$ is the product of distinct primes, and find $m+nk$.

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