
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
Let \( ABC \) be an equilateral triangle with side length 1. Points \( D, E, F \) lie inside triangle \( ABC \) such that \( A, E, F \) are collinear, \( B, F, D \) are collinear, \( C, D, E \) are collinear, and triangle \( DEF \) is equilateral. Suppose that there exists a unique equilateral triangle \( XYZ \) with \( X \) on side \(\overline{BC}\), \( Y \) on side \(\overline{AB}\), and \( Z \) on side \(\overline{AC}\) such that \( D \) lies on side \(\overline{XZ}\), \( E \) lies on side \(\overline{YZ}\), and \( F \) lies on side \(\overline{XY}\). Compute \( AZ \).

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