If the locus of the focus of the variable parabola, which always touches the curve \( C_2\) : \(x^2=-y\) at \( (0,0) \), is \( C_1 \) : \( x^2 + y^2 = 2ay^4 \).

If \( C_1\) and \(C_2\) have same length of Latus Rectum and they have no other intersection point, find \(a\).

Credits: My Mathematics Teacher.

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