The diagram shows a uniform rod \(A B\), of length \(4 a\) and weight \(W\), resting in equilibrium with its end \(A\) on rough horizontal ground. The rod rests at \(C\) on the surface of a smooth cylinder whose axis is horizontal. The cylinder rests on the ground and is fixed to it. The rod is in a vertical plane perpendicular to the axis of the cylinder and is inclined at an angle \(\theta\) to the horizontal, where \(\cos \theta = \frac { 3 } { 5 }\). A particle of weight \(k W\) is attached to the rod at \(B\). Given that \(A C = 3 a\), show that the least possible value of the coefficient of friction \(\mu\) between the rod and the ground is \(\frac { 8 ( 2 k + 1 ) } { 13 k + 19 }\). Given that \(\mu = \frac { 9 } { 10 }\), find the set of values of \(k\) for which equilibrium is possible.