Let $S$ denote the locus of the mid-points of those chords of the parabola $y ^ { 2 } = x$, such that the area of the region enclosed between the parabola and the chord is $\frac { 4 } { 3 }$. Let $\mathcal { R }$ denote the region lying in the first quadrant, enclosed by the parabola $y ^ { 2 } = x$, the curve $S$, and the lines $x = 1$ and $x = 4$.

Then which of the following statements is (are) TRUE?

\begin{center}
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(A) & $( 4 , \sqrt { 3 } ) \in S$ \\
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(B) & $( 5 , \sqrt { 2 } ) \in S$ \\
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(C) & Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - 2 \sqrt { 3 }$ \\
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(D) & Area of $\mathcal { R }$ is $\frac { 14 } { 3 } - \sqrt { 3 }$ \\
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\end{tabular}
\end{center}