Integers \(a\), \(b\), \(c\), and \(d\), not necessarily distinct, are chosen independently and at random from \(0\) to \(2007\), inclusive. Find the probability that \(ad - bc\) is even. The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest terms. Please calculate the value of \(k + m\).

Let's think step by step and output the final answer within \boxed{}.
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