
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 $ABC$, $AB = 12$, $BC = 18$, and $CA = 22$. Let $E$ be a point in the interior of $\overline{BC}$. Let points $I_B$ and $I_C$ denote the incenters of triangles $ABE$ and $ACE$, respectively. The circumcircles of triangles $BI_BE$ and $CI_CE$ meet at distinct points $Q$ and $E$. Determine the maximum possible area of $\triangle BQC$ and express it in the form $d - e\sqrt {f}$, where $d$, $e$, and $f$ are positive integers and $f$ is not divisible by the square of any prime. Find $d + e + f$.

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