CGMO 2016 Problem 5

Define a sequence $\{a_n\}$ by$$S_1=1,\ S_{n+1}=\frac{(2+S_n)^2}{ 4+S_n} (n=1,\ 2,\ 3,\ \cdots).$$ Where $S_n$ the sum of first $n$ terms of sequence $\{a_n\}$ . For any positive integer $n$ ,prove that$$a_{n}\ge \frac{4}{\sqrt{9n+7}}.$$