
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
Six circles with radii equal to 1 are placed on a plane such that the distance between the centers of any two of them is greater than \( d \). What is the smallest \( d \) for which there exists a line that does not intersect any of the circles, and on each side of which lie three circles?

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