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Geometry Level 4

A circle $$C_0$$ with radius 1 touches both the axes and line $$L_1$$ that passes through point $$P(0,4)$$ and cuts the $$x$$-axis at $$(x_1, 0)$$.

Another circle $$C_1$$ is drawn touching the $$x$$-axis, line $$L_1$$ and another line $$L_2$$ that passes through point $$P$$ and cuts the $$x$$-axis at $$(x_2, 0)$$ and this process is repeated $$n$$ times.

Find $$\displaystyle \lim_{n \to \infty} \frac {x_n}{2^n}$$.

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