CN-SMO 2021 Problem 1

A sequence $\{a_n\}$ is defined recursively by $a_1=\frac{1}{2},$ and for $n\ge 2,$ $0<a_n\leq a_{n-1}$ and $$a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.$$ $(1)$ Determine the general formula of the sequence $\{a_n\};$ $(2)$ Let $S_n=a_1+\cdots+a_n.$ Prove that for $n\ge 1,$ $\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).$