\[x + (1 + k)y = 0\]

\[(1 - k)x + ky = 1 + k\]

\[(1 - k)x + ky = 1 + k\]

In the above system of equations, \(x = \frac{a}{b} \) and \(y = \frac{-c}{d}\), where a, b, c and d are positive integers. Find the value of the sum \(a + b + c +d +k\).

**Details and assumption:**

\(\frac{a}{b}\) and \(\frac{-c}{d}\) are in lowest term.

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