KMO 2022 Problem 1

Three sequences ${a_n},{b_n},{c_n}$ satisfy the following conditions.

• $a_1=2, b_1=4, c_1=5$
• $\forall n, a_{n+1}=b_n+\frac{1}{c_n}, b_{n+1}=c_n+\frac{1}{a_n}, c_{n+1}=a_n+\frac{1}{b_n}$

Prove that for all positive integers $n$ , $$ $$ $max(a_n,b_n,c_n)>\sqrt{2n+13}$ .