\(ABC\) is an equilateral triangle. \(DE\) is the tangent to the inscribed circle with \(D\) on \(AB\) and \(E\) on \(AC\). It is given that \(DE\) perpendicular to \(AC\) and \(AE=10\).The side length of \(ABC\) can be expressed as \(a+b\sqrt{c}\) where \(a\) and \(b\) are positive integers and \(c\) is square-free positive integer; then find \(a+b+c\).

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