IMO 2021 — Complete Shortlist Question 1

Let $n$ be an integer, and let $A$ be a subset of $\left\{0,1,2,3, \ldots, 5^{n}\right\}$ consisting of $4 n+2$ numbers. Prove that there exist $a, b, c \in A$ such that $a\lt b\lt c$ and $c+2 a\gt 3 b$.