In the quadrilateral \(OABC\), \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=\mathbf b\), and \(\overrightarrow{OC}=\mathbf c\). The point \(M\) lies on \(AC\) such that \(AM:MC=2:1\). The point \(M\) also lies on \(OB\) such that \(OM:MB=3:2\).

(i) Find \(\overrightarrow{AC}\) in terms of \(\mathbf a\) and \(\mathbf c\).

(ii) Find \(\overrightarrow{OM}\) in terms of \(\mathbf a\) and \(\mathbf c\).

(iii) Find \(\overrightarrow{OM}\) in terms of \(\mathbf b\).

(iv) Find \(5\mathbf a+10\mathbf c\) in terms of \(\mathbf b\).

(v) Find \(\overrightarrow{AB}\) in terms of \(\mathbf a\) and \(\mathbf c\), simplifying your answer.