
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]


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The bases of two equilateral triangles with side lengths \(a\) and \(3a\) lie on the same line. The triangles are situated on opposite sides of the line and do not share any common points. The distance between the nearest ends of their bases is \(2a\). Find the distance between the vertices of the triangles that do not lie on the given line.

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