
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 $XYZ$, $XY = 12$, $YZ = 18$, and $ZX = 20$. Let $E$ be a point on line segment $\overline{YZ}$. Define points $J_Y$ and $J_Z$ as the incenters of triangles $XYE$ and $XZE$, respectively. The circumcircles of triangles $YJ_YE$ and $ZJ_ZE$ intersect at two distinct points $Q$ and $E$. Determine the maximum possible area of $\triangle YZQ$ and express it in the form $p - q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime. What is $p + q + r$?

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