The diagram shows parts of the curves with equations \(y=4e^{-2x}\) and \(y=1+0.5\sin3x\). Point \(P\) is a point of intersection of the curves, and the shaded region is bounded by the two curves and the \(y\)-axis.

(a) Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x=-0.5\ln(0.25+0.125\sin3x)\). [1]

(b) Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(P\) correct to 4 significant figures. Use an initial value of 0.5 and give the result of each iteration to 6 significant figures. [3]

(c) Hence find the area of the shaded region. Give your answer correct to 2 significant figures. [4]