In the coordinate plane, let $\mathrm{O}(0,0)$ and $\mathrm{A}(0,1)$ be two points. Suppose two points $\mathrm{P}(p,0)$ and $\mathrm{Q}(q,0)$ on the $x$-axis satisfy both of the following conditions (i) and (ii).

\begin{itemize}
\item[(i)] $0 < p < 1$ and $p < q$
\item[(ii)] Let $\mathrm{M}$ be the midpoint of segment $\mathrm{AP}$; then $\angle \mathrm{OAP} = \angle \mathrm{PMQ}$
\end{itemize}

(1) Express $q$ in terms of $p$.

(2) Find the value of $p$ such that $q = \dfrac{1}{3}$.

(3) Let $S$ be the area of $\triangle \mathrm{OAP}$ and $T$ be the area of $\triangle \mathrm{PMQ}$. Find the range of $p$ such that $S > T$.