11. (20 points) Let $x, y, z > 0$, and $xyz = 8$.
(1) If $x + y = 2 \sqrt{\frac{xy}{xy + 8}}$;
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(2) Let $\lceil x \rceil$ denote the smallest integer not less than the real number $x$, and $\sum$ denote the cyclic sum, find the value of $\left\lceil \sum \frac{1}{\sqrt{1 + x}} \right\rceil$.