3.43: Vector geometry: intersection $Z$ of $OY$ and $AX$

In triangle $OAB$, $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$. $X$ is the mid-point of $OB$ and $Y$ is the mid-point of $AB$. Lines $OY$ and $AX$ meet at $Z$ with $\overrightarrow{AZ}=m\overrightarrow{AX}$ and $\overrightarrow{OZ}=n\overrightarrow{OY}$.

Find $\overrightarrow{OZ}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $m$.

Find $\overrightarrow{OZ}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $n$.

Find the values of $m$ and $n$.

Hence find $\overrightarrow{OZ}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ only.