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 \( \dfrac{4}{3} \). Let \( R \) 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?
