Let $X_{1},...,X_{n}$ be a random sample from $n(\mu ,\sigma ^{2})$ where $\mu$, $\sigma$ are both unknown and $n\geq 2$. Denote $U=\sum_{i=1}^{n}(X_i - \bar{X})^2$. Let $V=cU$ be an estimator of $\sigma ^{2}$ where c is a positive constant. I found the MSE to be
$$MSE (V)=\frac{2\sigma^4}{n+1}$$
Now, I need help with finding the estimator which has the smallest MSE among estimators of the forms $cU$. How do I go about it?