Steady-State Gain with Q=1, R=2

Consider the scalar local-level model in steady state: $x_t=x_{t-1}+w_t$, $w_t\sim N(0,1)$, and $y_t=x_t+v_t$, $v_t\sim N(0,2)$. Compute the steady-state posterior variance $C$ and the steady-state Kalman gain $K$.