Let \( z \) be a complex number such that \( z^{12} = 1 \) and \( z \neq 1 \). Compute the value of
\[
\frac{z + z^2}{z^3 - z^4} + \frac{z^5 - z^6}{z^7 + z^8} + \frac{z^9 + z^{10}}{z^{11} - 1}.
\] Let's think step by step and output the final answer within \boxed{}.
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