A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\frac{4}{3} m g\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between \(P\) and the surface is \(\frac{1}{3}\). The particle is projected along the surface in the direction \(O P\) with a speed of \(\frac{1}{2} \sqrt{g a}\).

Find the greatest extension of the string during the subsequent motion.