Not as They Seem to Be

Geometry Level 4

Triangle \(ABC\) is isosceles, with \(AB=AC\) and altitude \(AM=11.\) Suppose that there is a point \(D\) on \(\overline{AM}\) with \(AD=10\) and \(\angle BDC=3\angle BAC.\) Then the perimeter of \(\triangle ABC\) may be written in the form \(a+\sqrt{b},\) where \(a\) and \(b\) are integers. Find \(a+b.\)

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