Derive the MSE-optimal variance divisor

For $n$ independent normal draws, consider estimating the variance by $\hat\sigma^2 = c\sum_i (X_i - \bar X)^2 = c\,\text{SS}$ for a constant $c$.

Using $\mathbb{E}[\text{SS}] = (n-1)\sigma^2$ and $\text{Var}(\text{SS}) = 2(n-1)\sigma^4$, find the $c$ that minimizes MSE and show it corresponds to dividing by $n+1$.