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Solve $\frac{{\partial}^{2} z}{\partial x \partial y} = {x}^{2} y$ given the conditions $z(x,0)={x}^{2}$ and $z(1,y) = \cos {y}$.

The solution is of the form $z(x,y) = A{x}^{3} {y}^{2} + B \cos {y} - C{y}^{2} +D{x}^{2} -1.$ What is $A + B + C + D$?

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