4. If the complex number $z$ satisfies $\left(\frac{z-\sqrt{3}}{z+\sqrt{3}}\right)^{3}<0$, then the minimum value of $|z+7|$ is $\qquad$

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4. $5 \sqrt{2}-2$.

Let the principal value of the argument of $\frac{z-\sqrt{3}}{z+\sqrt{3}}$ be $\theta(\theta \in[0,2 \pi))$.
Since $\left(\frac{z-\sqrt{3}}{z+\sqrt{3}}\right)^{3}7-\sqrt{3}>5 \sqrt{2}-2 .
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