Embarrassing to the Solver

Geometry Level 5

In triangle \(ABC\), the angle bisector from \(A\) and the perpendicular bisector of \(BC\) meet at point \(D\), the angle bisector from \(B\) and the perpendicular bisector of \(AC\) meet at point \(E\), and the perpendicular bisectors of \(BC\) and \(AC\) meet at point F. Given that \(\angle ADF = 5\) , \(\angle BEF = 10\) , and \(AC = 3\), let the length of \(DF\) be \(\sqrt{a}\) where \(a\) is a square-free positive integer. Find \(a\).

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