$ABCD$ is a cyclic quadrilateral. If $\angle B = \angle D$, $AC \cap BD = \{E\}$, $\angle BCD = 150^\circ$, $|BE| = x$, $|AC| = z$, then find $|ED|$ in terms of $x$ and $z$. The original answer is in the form \(\frac{kx + mz}{n}\), where n cannot be divisible by 2. Please provide the value of \(k + m + n\).

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