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Express $\vec{OB}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

Express $\vec{OM}$ and $\vec{MN}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

Show that $\vec{MN}$ can be written in the form $k(\mathbf{a} - 2\mathbf{c})$, stating the value of scalar $k$.

Point $P$ lies on $AC$ such that $\vec{AP} = \frac{2}{5}\vec{AC}$. Deduce whether $M, N$ and $P$ are collinear.