A ray of light passing through the point \( P(2, 3) \) reflects on the \( x \)-axis at point \( A \) and the reflected ray passes through the point \( Q(5, 4) \). Let \( R \) be the point that divides the line segment \( AQ \) internally into the ratio \( 2\!:\!1 \). Let the co-ordinates of the foot of the perpendicular \( M \) from \( R \) on the bisector of the angle \( PAQ \) be \( (\alpha, \beta) \). Then, the value of \( 7\alpha + 3\beta \) is equal to ________.
