On an Argand diagram shade the region whose points represent complex numbers \(z\) which satisfy both the inequalities \(| z - 4 - 3 i | \leqslant 2\) and \(\arg ( z - 2 - i ) \geqslant \frac { 1 } { 3 } \pi\).
Calculate the greatest value of \(\arg z\) for points in this region.