The diagram shows the part of the curve \(y = x^2 \cos 3x\) for \(0 \leq x \leq \frac{1}{6}\pi\), and its maximum point \(M\), where \(x = a\).

(a) Show that \(a\) satisfies the equation \(a = \frac{1}{3} \arctan \left( \frac{2}{3a} \right)\).

(b) Use an iterative formula based on the equation in (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.