# Concentric Circles, Tangent Circles

Geometry Level 5

Circle $$\Gamma_1$$ with center $$O_1$$ is drawn. A smaller circle $$\Gamma_2$$ with center $$O_1$$ is drawn. Circle $$\Gamma_3$$ with center $$O_2$$ is drawn such that it is internally tangent to both circles $$\Gamma_1$$ and $$\Gamma_2$$. The radius of circle $$\Gamma_1$$ that is perpendicular to $$\overline{O_1O_2}$$ is drawn; it intersects $$\Gamma_1$$ at $$A$$, $$\Gamma_2$$ at $$B$$, and $$\Gamma_3$$ at $$P$$.

Let the minimum possible value of $$\dfrac{AP}{AB}$$ for any choice of circles $$\Gamma_1$$ and $$\Gamma_2$$ be $$m$$. Find $\lfloor 1000m\rfloor$

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