An object consists of a uniform lamina with a particle attached. The lamina \(ABCEFD\), of mass \(m\), is formed from a rectangle \(ABCD\) and an isosceles triangle \(CEF\), where \(F\) is the midpoint of \(CD\). The rectangle has sides \(AB=2a\) and \(AD=a\). The triangle has base \(a\) and height \(2a\). A particle of mass \(km\) is attached at \(E\). The object rests with edge \(AD\) on horizontal ground and is on the point of toppling about \(D\). Find \(k\).