Let \(X\) be a single observation from a discrete distribution with probability distribution function

\[f(x \mid \theta)= \begin{cases}n^{-1} \theta & x \in\{1,2, \ldots, n\} \\ 1-\theta & x=0 \\ 0 & \text { elsewhere }\end{cases}\]

where \(\theta \in \Omega=\left\{(n+1)^{-1}, 2(n+1)^{-1}, \ldots, n(n+1)^{-1}ight\}\). Suppose that the prior distribution on \(\theta\) is a \(\operatorname{Uniform}\left\{(n+1)^{-1}, 2(n+1)^{-1}, \ldots, n(n+ight.\) \(1)^{-1}\) \} distribution. Suppose that \(X=x\) is observed. Compute the posterior distribution of \(\theta\) and the Bayes estimator of \(\theta\) using the squared error loss function.