Estimation with Equicorrelated Normal Samples

You have $n$ observations $X_1, \ldots, X_n$ drawn from a $N(\mu, 1)$ distribution, but they are not independent -- every pair has the same correlation $r$, so the covariance matrix is: $\Sigma = (1-r)I + r\mathbf{1}\mathbf{1}^\top$ where $\mathbf{1}$ is the all-ones vector. (a) How would you estimate $\mu$? Is the sample mean $\bar{X}$ still the best linear unbiased estimator? (b) Derive the test statistic for testing $H_0: \mu = \mu_0$ against $H_1: \mu \neq \mu_0$. How does correlation affect the power of this test? (c) What is the valid range of $r$ for this covariance structure to be well-defined?