Consider triangle \( \triangle ABC \) with vertices at \( A(0, 0) \), \( B(6, 0) \), and \( C(4, 3) \). Let \( H \) be the orthocenter of \( \triangle ABC \), and \( G \) be the centroid of \( \triangle ABC \). Determine the coordinates of the fourth vertex \( D \) such that quadrilateral \( ABDC \) is a parallelogram with integer side lengths. Then, calculate the area of quadrilateral \( ABDC \). Let's think step by step and output the final answer within \boxed{}.
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