A parabola \(y=x^2\) is drawn. An ellipse is drawn with one focus at the focus of the parabola and another focus on the line \(y=2014\). Given that the ellipse is tangent to the parabola at coordinate \((a,a^2)\), then find the maximum possible integer value of \(a\).
Details and Assumptions
- The diagram above is not the correct solution (I just have trouble graphing tilted ellipses to my liking).