
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 \(B_1C_1\) of the equilateral triangles \(ABC\) and \(A_1B_1C_1\) coincide (the vertices of both triangles are listed in a clockwise order). Find the angle between the lines \(AA_1\) and \(BB_1\), and also the ratio of the lengths of the segments \(AA_1\) and \(BB_1\).

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