Convex Quadratic?

Geometry Level 3

Consider a curve \(y=ax^2 +bx+c\) with \( a, b, c \in \mathbb{N} \) which passes through four points \(A(-2,3),B(-1,1),C(\alpha ,\beta ),D(2,7)\).

All these points are taken in given order for constructing a convex quadrilateral, which has maximum possible area.

Find minimum possible value of \(a+b+c+2\alpha +4\beta.\)

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