The odd theoretical ant

Algebra Level 3

An ant travels along a straight path with a distance of xx meters. From then on, it turns left and covers 23 \frac{2}{3} of the straight distance it traveled before turning, and continues doing this until it eventually reaches an unknown point PP (referred to as the red point in the middle of the picture). Refer to the figure above.

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

Challenge: Determine an explicit formula for kk.

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