If \( z_{1} \) and \( z_{2} \) both satisfy the relation \( z+\bar{...

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If \( z_{1} \) and \( z_{2} \) both satisfy the relation \( z+\bar{z}=2|z-1| \) and \( \arg \left(z_{1}-z_{2}\right)=\frac{\pi}{4} \), find the imaginary part of \( \left(z_{1}+z_{2}\right) \).
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