Three Concentric Circles

Geometry Level 4

Three concentric circles \(A,B,C\) have the property that the radius of \(A\) is bigger than the radius of \(B\), and the radius of \(B\) is bigger than the radius of \(C\). A chord of circle \(B\) of length \(2\) is drawn such that it is tangent to circle \(C\); a chord of circle \(A\) of length \(x\) is drawn such that it is tangent to circle \(B\). If the area between circle \(A\) and \(B\) is \(2014\) times the area between circle \(B\) and circle \(C\), then the value of \(x\) can be expressed as \(\sqrt{a}\) for some positive integer \(a\). Find the last three digits of \(a\).

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