Let \(\triangle ABC\) be a triangle with \(BC= 5, CA= 6, AB= 7.\) Let \(D, E\) be the midpoints of \(AB, AC\) respectively. The circumcircles of \(\triangle BCD\) and \(\triangle ADE\) meet for a second time at \(X\) (\(X \neq A\)). Given that \(AX^2= \dfrac{a}{b}\) for some coprime positive integers \(a, b,\) find the last three digits of \(a+b.\)

**Details and assumptions**

- This problem is inspired by a recent problem that appeared in the Chinese Girls Mathematical Olympiad 2014.

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