Let \(\mathbf{t}\) be the treatment assignment vector, and let \(P_{W}\) be the \(W\)-orthogonal projection onto the subspace \(\operatorname{span}(\mathbf{1}, \mathbf{t})\). Show that the transformation

\[
N_{n}(\mu, \Sigma) \stackrel{g}{\longmapsto} N_{n}\left(\mu+g_{0} \mathbf{t}, Q_{\mathcal{X}} \Sigma+e^{g_{1}} P_{W} \Sigmaight)
\]

is an action of the additive group \(\mathbb{R}^{2}\) on the space of Gaussian distributions. Describe the orbit of the distribution \(N_{n}\left(\mathbf{1}, I_{n}ight)\).