A light elastic string has natural length \(8a\) and modulus of elasticity \(5mg\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\), which are a distance \(12a\) apart on a smooth horizontal table.

The particle \(P\) is held on the table so that \(AP=BP=L\). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(AB\), it has speed \(\sqrt{80ag}\).

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

(b) Find the initial acceleration of \(P\) in terms of \(g\).