TUYMAADA 2020 S2

Given positive real numbers $a_1, a_2, \dots, a_n$ . Let $$m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right).$$ Prove the inequality $$\sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m.$$