Integral octagon

We will say that an octagon is integral if it is equiangular, its vertices are lattice points (i.e., points with integer coordinates), and its area is an integer.

For example, the figure on the above shows an integral octagon of area 19.

Determine the smallest positive integer \(K\) so that there exists an integral octagon, none of whose sides are parallel to the coordinate axes, that has an area of \(K\).

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