NMC 2012 Problem 3

Find the smallest positive integer $n$, such that there exist $n$ integers $x_{1}, x_{2}, \ldots, x_{n}$ (not necessarily different), with $1 \leq x_{k} \leq n, 1 \leq k \leq n$, and such that

$$x_{1}+x_{2}+\cdots+x_{n}=\frac{n(n+1)}{2}, \text { and } x_{1} x_{2} \cdots x_{n}=n!$$

but $\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \neq\{1,2, \ldots, n\}$.