
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 \( ABC \), \( BC = 30 \) and \( \angle C = 45^\circ \). Let the perpendicular bisector of \( BC \) intersect \( BC \) at \( D \) and extended \( AC \) at \( E \). Suppose \( F \) is the midpoint of \( AC \). Find the lengths of \( DE \) and \( EF \).

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