A robot moves through a one-dimensional state space $X=\mathbb{R}$ by choosing actions from a one-dimensional action space $U=\mathbb{R}$. The state transition function is \[ x_{k+1} = x_k + 2u_k + 3\theta_k, \] in which the noise value $\theta_k$ is selected randomly from a Gaussian distribution with mean $0$ and variance $0.5$. At each time step, the robot receives an observation from a one-dimensional observation space $Y=\mathbb{R}$, according to the observation function \[ y_k = x_k + 4\psi_k,\] in which the noise value $\psi_k$ is selected randomly from a Gaussian distribution with mean $0$ and variance $1.5$.

At a certain time step $k$, the robot knows that its most likely state is $\mu_k = 5$, with variance $\Sigma_k=0.3$. The robot receives an observation $y_k = 7$ and executes an action $u_k=0.5$.

Using the Kalman filter, compute the most likely value $\mu_{k+1}$ of $x_{k+1}$ and the variance $\Sigma_{k+1}$ of that estimate. Show your work.

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