Travelling Ant

Geometry Level 2

An ant finds itself trapped in the xyxy-plane, and its initial position is (1,0).(1,0).

Let SkS_k denote the circle with radius kk centered around the origin. Starting from (1,0)(1,0), the ant walks 1 unit counter-clockwise on S1.S_{1}. Then, it walks directly (radially outward) to S2,S_2, on which it will walk 2 units counter-clockwise. Then, it will walk directly to S3S_3 and walk 3 units counter-clockwise, and so, with the ant walking kk units on Sk.S_k. (See the image above.)

When the ant crosses the positive xx-axis for the first time since it left (1,0)(1,0), it is on SnS_{n}. What is nn?

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