
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 $ZX = 22$. Let $E$ be a point in the interior of $\overline{YZ}$. Let points $I_Y$ and $I_Z$ denote the incenters of triangles $XYE$ and $XZE$, respectively. The circumcircles of triangles $YI_YE$ and $ZI_ZE$ meet at distinct points $Q$ and $E$. The maximum possible area of $\triangle YQZ$ can be expressed in the form $a - b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.

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