Proving that $ \prod_{i=1}^{n} (1 + a_{i}) \ge \left(1 + \prod_{i=1}^{n} a_{i}^{1/n} \right)^{n} $

Question

I'd like some help on proving the following inequality $$ \prod_{i=1}^{n}\left(1+a_{i}\right) \ge \bigg(1 + \prod_{i=1}^{n} a_{i}^{1/n}\bigg)^{n}, $$ given that $ a_{i} > 0\,\, \forall\, i\in\{1,2,\cdots,n\} $.

I tried to use AM-GM inequality on the right-hand side, but without success. Any help would be appreciated.