
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 an equilateral triangle \( ABC \) with side length 1, points \( D \) and \( E \) are taken on sides \( AB \) and \( AC \) respectively such that when the triangle is folded along line segment \( DE \), vertex \( A \) falls exactly on side \( BC \). The minimum length of \( AD \) is \( \qquad \)

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