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Let $c$ be the smallest positive real number such that for all positive integers $n$ and all positive real numbers $x_{1},....,x_{n}$ , the below inequality holds. Compute $\lceil{2022c}\rceil$

$$\displaystyle{\sum_{k=0}^{n}\frac{(n^3+k^3-k^2n)^{\frac{3}{2}}}{\sqrt{{x_1}^{2}+\cdots+{x_k}^{2}+x_{k+1}+\cdots+x_{n}}}\le{\sqrt{3}\left(\sum_{i=1}^{n}\frac{i^{3}(4n-3i+100)}{x_{i}}\right)+cn^5+100n^4}}$$



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