
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 side lengths \( AB=3 \), \( BC=4 \), and \( CA=5 \). Points \( D \) and \( E \) are on ray \( AB \) with \( AB<AD<AE \). The point \( F \neq C \) is a point of intersection of the circumcircles of \( \triangle ACD \) and \( \triangle EBC \) satisfying \( DF=3 \) and \( EF=8 \). Determine \( BE \).

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