A light elastic string has natural length \(a\) and modulus of elasticity \(12 m g\). One end of the string is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle hangs in equilibrium vertically below \(O\). The particle is pulled vertically down and released from rest with the extension of the string equal to \(e\), where \(e\gt \frac{1}{3} a\). In the subsequent motion the particle has speed \(\sqrt{2 g a}\) when it has ascended a distance \(\frac{1}{3} a\).

Find \(e\) in terms of \(a\).