
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 $ABC$, the side lengths are $AB=7$, $BC=8$, and $CA=9$. Points $D$ and $E$ are on ray $AB$ such that $AB<AD<AE$. Point $F \neq C$ is on the intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$, with $DF=3$ and $EF=8$. Determine the length of $BE$, which can be expressed in the form $\tfrac{p+q\sqrt{r}}{s}$, where $p$, $q$, $r$, and $s$ are positive integers, $p$ and $s$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $p+q+r+s$.

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