
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
Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \) with edge length \( a \). The ends of a line segment, forming a \( 60^\circ \) angle with the plane of the face \( A B C D \), lie on the lines \( A B_{1} \) and \( B C_{1} \). What is the minimum length this line segment can have?

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