The Crazy Ant

Calculus Level 5

A crazy ant is standing on the origin. It begins by walking 1 unit in the +x+x-direction and then turns 4545^\circ counterclockwise and walks 12\frac{1}{2} units in that direction. The ant then turns another 4545^\circ and walks 13\frac{1}{3} units in that direction. The ant keeps doing this endlessly. How far is the ant's final position from its initial position?

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