
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 congruent copies of the parabola \( y = x^2 \) are arranged in the plane so that each vertex is tangent to a circle. Each parabola is tangent to its two neighbors. This time, however, instead of a 60-degree angle, arrange the parabolas so that each is tangent to a line inclined at a 45-degree angle. Find the radius of the circle under this new configuration.

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