Cursed angle bisectors!

Geometry Level 5

Let there exist a right-angled triangle \(ABC\), with \(\angle ABC = 90^{\circ}\). Let \(CD\) be the internal angle bisector of \(\angle ACB\), with \(D\) on \(AB\) and let \(AE\) be the internal angle bisector of \(\angle BAC\), with \(E\) on \(BC\).

Let the length of \(AE\) is \(9\) and the length of \(CD\) is \(8\sqrt{2}\). If the length of \(AC\) is \(a \sqrt{b}\) where \(a\) and \(b\) are positive integers and \(b\) is square-free, find the value of \(a+b\).

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