
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 triangle is folded along line segment \(DE\), vertex \(A\) falls precisely on side \(BC\). The minimum length of \(AD\) is \(\_\_\_\_\_\_\_ \).

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