Tangents and Triangles

Geometry Level 4

The tangent at \(A\) of the circumcircle of the triangle \(ABC\) is drawn. The points \(D\) and \(E\) are constructed on it such that \(BD\) is parallel to \(CA\) and that \(CE\) is parallel to \(BA\). The lines \(BD\) and \(CE\) intersect the circle \(ABC\) again at \(X\) and \(Y\) respectively.

Given that \(BC = 10\), find \(AX + AY\).

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