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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