One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(k\), is attached to a particle \(P\) of mass \(m\). The other end of the string is attached to a fixed point \(Q\). The particle \(P\) is projected vertically upwards from \(Q\). When \(P\) is moving upwards and at a distance \(\frac{4}{3} a\) directly above \(Q\), it has a speed \(\sqrt{2 g a}\). At this point, its acceleration is \(\frac{7}{3} g\) downwards.

Show that \(k=4 m g\) and find in terms of \(a\) the greatest height above \(Q\) reached by \(P\).