A particle \(P\) is projected from a point \(O\) on a horizontal plane and moves freely under gravity. The initial velocity of \(P\) is \(25 \mathrm{~ms}^{-1}\) at an angle \(\theta\) above the horizontal, where \(\tan \theta=\frac{4}{3}\). At point \(A\), the direction of motion of \(P\) makes an angle of \(45^{\circ}\) with the downward vertical through \(A\). (a) By differentiating the equation of the trajectory or otherwise, find the coordinates of \(A\).

At point \(A\), the particle strikes a fixed smooth barrier, rebounds, and lands on the horizontal plane. The barrier is inclined at an angle of \(45^{\circ}\) to the horizontal. (b) Find the speed of \(P\) immediately before it collides with the barrier. (c) Given that the coefficient of restitution between the barrier and the particle is \(\frac{1}{9}\), find the horizontal distance travelled by \(P\) after it strikes the barrier.