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).

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