# Brilli's Spiral Journey

Algebra Level 4

Brilli the Ant starts at the origin and travels in a spiral pattern as follows:

First, he travels along the positive x-axis a distance of $$1$$. Then, he rotates $$60^\circ$$ counterclockwise and travels a distance of $$\frac{1}{2}$$. Then, he rotates $$60^\circ$$ counterclockwise and travels a distance of $$\frac{1}{4}$$. For each subsequent movement, he rotates $$60^\circ$$ counterclockwise and travels half as far as his last movement.

As Brilli continues indefinitely on this path, he approaches point $$M$$.

If the distance from the origin to point $$M$$ is $$\sqrt{\frac{a}{b}}$$, where $$a$$ and $$b$$ are positive integers such that $$\gcd(a,b)=1$$, then what is $$a+b$$?

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