
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
A regular hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ and an equilateral triangle $A_{1} A_{3} A_{5}$ are inscribed in a circle of radius $r$. From each vertex of the equilateral triangle, we measure a distance $x$ along its sides in both directions. Connecting the endpoints of these distances in the same direction forms a hexagon.

1. For what value of $x$ will the resulting hexagon be regular?

2. Determine the areas of the two hexagons and the equilateral triangle.

3. Find the ratio of the radii of the circles circumscribed around the two regular hexagons.

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