
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 $D$ be a point in the interior of $\overline{BC}$. Let points $J_B$ and $J_C$ denote the incenters of triangles $ABD$ and $ACD$, respectively. The circumcircles of triangles $BJ_BD$ and $CJ_CD$ meet at distinct points $Q$ and $D$. The maximum possible area of $\triangle BQC$ can be expressed 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. Find $p + q + r$.

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