Let the line \( L:\ \sqrt{2}x+y=\alpha \) pass through the point of the intersection \( P \) (in the first quadrant) of the circle \( x^2+y^2=3 \) and the parabola \( x^2=2y \). Let the line \( L \) touch two circles \( C_1 \) and \( C_2 \) of equal radius \( 2\sqrt{3} \). If the centres \( Q_1 \) and \( Q_2 \) of the circles \( C_1 \) and \( C_2 \) lie on the y-axis, then the square of the area of the triangle \( PQ_1Q_2 \) is equal to ______.
