The odd theoretical ant

Algebra Level 3

An ant travels along a straight path with a distance of \(x\) meters. From then on, it turns left and covers \( \frac{2}{3}\) of the straight distance it traveled before turning, and continues doing this until it eventually reaches an unknown point \(P\) (referred to as the red point in the middle of the picture). Refer to the figure above.

The distance of \(P\) from the ant's starting point is \(kx\) meters, where \(k\) is a positive constant which can be expressed in the form \( \frac{a \sqrt {b}}{c} \), where \( \gcd(a,c) = 1\) and \(b\) is square free. Determine the value of \( a + b + c \).

Challenge: Determine an explicit formula for \(k\).

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