
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
Let  $P_1$ ,  $P_2$ ,  $P_3$ ,  $P_4$ ,  $P_5$ , and  $P_6$  be six parabolas in the plane, each congruent to the parabola  $y = x^2/16$ .  The vertices of the six parabolas are evenly spaced around a circle.  The parabolas open outward with their axes being extensions of six of the circle's radii.  Parabola  $P_1$  is tangent to  $P_2$ , which is tangent to  $P_3$ , which is tangent to  $P_4$ , which is tangent to  $P_5$ , which is tangent to  $P_6$ , which is tangent to  $P_1$ .  What is the diameter of the circle?

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