The least value of $|z|$ where $z$ is complex number which satisfies the inequality
$$
\exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|} \log _e 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{5}+9 i|, i=\sqrt{-1}
$$
, is equal to:
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