
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
Four distinct circles \( C, C_{1}, C_{2}, C_{3} \) and a line \( L \) are given in the plane such that \( C \) and \( L \) are disjoint and each of the circles \( C_{1}, C_{2}, C_{3} \) touches the other two, as well as \( C \) and \( L \). Assuming the radius of \( C \) to be 1, determine the distance between its center and \( L \).

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