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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