
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
Consider a triangle $ABC$ with sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$ of length 26, 15, and 37, respectively. Let $\omega$ be the circle circumscribed around $\triangle ABC$. Let $E$ be the intersection of $\omega$ and the perpendicular bisector of $\overline{BC}$ not on the same side of $\overline{BC}$ as $A$. Find the length of $\overline{BE}$ and express it as $p\sqrt{q}$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Determine the greatest integer less than or equal to $p + \sqrt{q}$.

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