A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C=3 a\) and \(D B=a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac{3}{4} m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k\omega\). \(AC\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram).

Find the value of \(k\).