
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
Given that in the right triangle \( \triangle ABC \), it is known that \( AC = 6 \), point \( E \) is on side \( AC \) and satisfies \( CE = 2AE \), point \( D \) is the midpoint of the hypotenuse \( AB \), and point \( F \) is a moving point on segment \( BC \) and satisfies \( \angle EDF = 90^\circ \), determine the maximum value of \( BF - FC \).

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