
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
Triangle $ABC$ has sides $AB=3$, $BC=4$, and $CA=5$. Points $D$ and $E$ are placed on ray $AB$ so that $AB<AD<AE$. Point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ABD$ and $\triangle EBC$ such that $DF=3$ and $EF=4$. Find the length of $BE$, express it as $\frac{p+q\sqrt{r}}{s}$, where $p$, $q$, $r$, and $s$ are positive integers with $p$ and $s$ relatively prime, and $r$ is a prime number.

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