The diagram shows the trapezium \(O A B C\), where \(\overrightarrow{O A}=4 \mathbf{a}, \overrightarrow{O C}=\mathbf{c}\), and \(\overrightarrow{C B}=2 \mathbf{a}\). The point \(D\) lies on \(A B\) such that \(A D: D B=2: 1\). The point \(X\) is the point of intersection of the lines \(O D\) and \(A C\). It is given that \(\overrightarrow{A X}=\lambda \overrightarrow{A C}\) and \(\overrightarrow{O X}=\mu \overrightarrow{O D}\).

Find in terms of \(\mathbf{a}\) and \(\mathbf{c}\) (a) \(\overrightarrow{A B}\)

(b) \(\overrightarrow{O D}\).

(c) Find \(\overrightarrow{O X}\) in terms of \(\mathbf{a}, \mathbf{c}\) and \(\mu\).

(d) Find \(\overrightarrow{A X}\) in terms of \(\mathbf{a}, \mathbf{c}\) and \(\lambda\).

(e) Hence find the values of \(\lambda\) and \(\mu\).