Geometry Level 2

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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