
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 side of the rhombus \(ABCD\) is \(a\), and the acute angle \(ABC\) is \(a\). Equilateral triangles are constructed externally on the segments \(AD\) and \(BC\). Find the distance between their centers.

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