The curves \(C_{1}: y=\cosh x\) and \(C_{2}: y=\sinh 2 x\) intersect at the point where \(x=a\).
(a) Find the exact value of \(a\), giving your answer in logarithmic form.
(b) Sketch \(C_{1}\) and \(C_{2}\) on the same diagram.
(c) Find the exact value of the length of the \(\operatorname{arc}\) of \(C_{1}\) from \(x=0\) to \(x=a\).