Let \( z_{1}, z_{2}, z_{3} \) be vertices of an equilateral triangle with \( z_{0} \) its centro...

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Let \( z_{1}, z_{2}, z_{3} \) be vertices of an equilateral triangle with \( z_{0} \) its centroid. If \( z_{1}^{2}+z_{2}{ }^{2}+z_{3}{ }^{2}=k\left|z_{0}\right|^{2} \), then \( k= \)
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