
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
Equilateral triangle \( \triangle ABC \) has side length 6. Let \( \omega \) be the circle through \( A \) and \( B \) such that \( CA \) and \( CB \) are both tangent to \( \omega \). A point \( D \) on \( \omega \) satisfies \( CD = 4 \). Let \( E \) be the intersection of line \( CD \) with segment \( AB \). What is the length of segment \( DE \)?

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