The straight line \(y=2 x+1\) intersects the curve \(y+x y+3 x^{2}=15\) at the points \(A\) and \(B\). The point \(C\) with coordinates \(\left(\frac{21}{10}, k\right)\) lies on the perpendicular bisector of \(A B\). (a) Find the exact value of \(k\).

(b) The point \(D\) lies on the perpendicular bisector of \(A B\) such that its perpendicular distance from \(A B\) is twice that of the point \(C\) from \(A B\). Find the possible coordinates of \(D\).