HMMT 2020 U32

Find the smallest real constant $\alpha$ such that for all positive integers $n$ and real numbers $0=y_{0}<$ $y_{1}<\cdots<y_{n}$, the following inequality holds:

$$\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}} \text {. }$$