I would like to construct a sequence of discrete random variable $X_2, X_3,...,X_n,...$, where $X_n \in\{0,1,2,...,n-1\}$. Given any $\epsilon \in (0,1)$, its Shannon entropy and min-entropy should satisfy the following relationships

\begin{cases} H(X_n)\geq(1-\epsilon)\log_2(n)\\ H_{min}(X_n)=const \end{cases} for all $n\geq\mathbb{N}_{\epsilon}$ and some $const > 0$.

My understanding is that the Shannon entropy indicates the underlying distribution should be approximately uniform. And the min-entropy suggests that the largest possibility of $X_n$ should be $2^{-const}$. But I am stuck with coming up with such a distribution. Is there anyone who could provide some hints?