
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
The midpoints of the sides \(BC\) and \(B_{1}C_{1}\) of equilateral triangles \(ABC\) and \(A_{1}B_{1}C_{1}\) coincide (the vertices of both triangles are listed in clockwise order). Find the measure of the angle between lines \(A A_{1}\) and \(B B_{1}\), as well as the ratio of the lengths of segments \(A A_{1}\) and \(B B_{1}\).

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