Suppose the single random variable $X$ has a uniform distribution on the interval $[0, \theta]$ and it is required to estimate $\theta$ with the loss function

$$L(\theta, a)=c(\theta-a)^{2}$$

where $c>0$.

Find the posterior distribution for $\theta$ and the optimal Bayes point estimate with respect to the prior distribution with density $p(\theta)=\theta e^{-\theta}, \theta>0$.