
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
An equilateral triangle is inscribed in a circle of radius \( r \). Then, a new equilateral triangle is formed by joining the midpoints of the sides of the first triangle inscribed within it. This process is repeated indefinitely to form a sequence of nested equilateral triangles. What is the limit of the sum of the perimeters of all the triangles thus drawn?
A) \( 3r\sqrt{3} \)
B) \( 6r \)
C) \( 6r\sqrt{3} \)
D) \( 9r\sqrt{3} \)

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