
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
Seven congruent copies of the parabola $y = x^2$ are arranged uniformly in a circular manner so that each vertex is tangent to a circle, and each parabola is tangent to its two neighbors. Find the radius of the circle.

[asy]
unitsize(1 cm);

real func (real x) {
  return (x^2 + 2/3);
}

path parab = graph(func,-1.5,1.5);

draw(parab);
draw(rotate(51.43)*(parab));
draw(rotate(102.86)*(parab));
draw(rotate(154.29)*(parab));
draw(rotate(205.71)*(parab));
draw(rotate(257.14)*(parab));
draw(rotate(308.57)*(parab));
draw(Circle((0,0),2/3));
[/asy]

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