
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
All edges of a regular triangular prism \( ABC A_1 B_1 C_1 \) have length \( a \). Points \( M \) and \( N \) lie on the lines \( BC_1 \) and \( CA_1 \), respectively, such that the line \( MN \) is parallel to the plane \( AA_1 B \). What is the minimum length of the segment \( MN \)?

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