Double inequality with a certain number of reals

Question

I've encountered the following problem that I don't know how to solve:
Given positive natural $n$ and positive real $x_1, x_2, ..., x_n$ prove that there exists such positive natural $N$ that
$(1+\frac1n)^N\ge 2 (x_1+x_2+...+x_{n-1}) + \frac{N}nx_n\ge N^n$.
I don't even know how to start with either of the parts, leave alone both of them simultaneously. I supposed it may be done by the inequality of means but my attempts didn't show anything interesting.
Appreciate your help.