
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 \( A B C A_{1} B_{1} C_{1} \) have a length \( a \). Consider segments with their ends on the diagonals \( B C_{1} \) and \( C A_{1} \) of the lateral faces, which are parallel to the plane \( A B B_{1} A_{1} \). Find the minimum length of such segments.

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