
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 has a side length of $b$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. The midpoints are determined such that each side of the second triangle is $\frac{1}{3}$ the length of the sides of the first triangle. This process is repeated indefinitely, always forming a new triangle by joining the midpoints of the sides of the previous one to get sides $\frac{1}{3}$ of the previous triangle's sides. Calculate the limit of the sum of the perimeters of all the triangles thus drawn.
**A)** $\frac{3b}{2}$  
**B)** $\frac{6b}{5}$  
**C)** $\frac{9b}{2}$  
**D)** $\frac{4b}{3}$

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