
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 $ABC$, $AB = 13$, $BC = 15$, and $CA = 14$. Let $E$ be a point on 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 as $a - b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers with $c$ not divisible by the square of any prime.

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