I need help finding the limit of this sequence

I need help with the following exercise that reads as follows

Calculate the limit of the sequence defined by:

$$x_n=\frac{1}{n}(n+\frac{n-1}{2}+\frac{n-2}{3}+...+\frac{2}{n-1}+\frac{1}{n}-\log(n!))$$

The idea is to find this limit by means of asymptotic equivalences, for this purpose consider the following

As $\log(n!)∼n\log(n)$ then $x_n ∼ \frac{1}{n}(n+\frac{n-1}{2}+\frac{n-2}{3}+...+\frac{2}{n-1}+\frac{1}{n}-n\log(n))$

I thought about establishing this equivalence, however I do not know how to arrive at the value of the limit.

Thank you for your help

Hint. Note that $$n+\frac{n-1}{2}+\frac{n-2}{3}+...+\frac{2}{n-1}+\frac{1}{n}=\sum_{k=1}^n\frac{n+1-k}{k}=(n+1)H_n-n$$ where $H_n=\sum_{k=1}^n\frac{1}{k}$ is the $n$-th harmonic number.

Recall that $H_n=\log(n)+\gamma +o(1)$. Moreover, use Stirling approximation for $n!$.