Inequality $(1+a_2)^{2}(1+a_3)^{3}... (1+a_n)^{n}>n^{n}$

Inequality $(1+a_2)^{2}(1+a_3)^{3}... (1+a_n)^{n}>n^{n}$

I need help proving this inequality.
Let $a_2,a_3,...,a_n$ be positive non zero real numbers such that their
product is $1$
Prove that
$$(1+a_2)^{2}(1+a_3)^{3}... (1+a_n)^{n}>n^{n}$$
I tried Bernoulli that yields a LHS greater than $n! $
I also tried a trig $tan^{2}$ substitution without success...
Thanks for helping.