TUYMAADA 2022 S4

For every positive $a_1, a_2, \dots, a_6$ , prove the inequality $$\sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2$$