Please help with this provide step by step solution Problem $3(40$ points$)$:

Consider a two$-$bit register. The register has four possible states: $00,01,10$ and $11.$ Initially, at time $t=0,$ the contents of the register is chosen at random to be one of these four states, each with equal probability. At each time step, the register is randomly manipulated as follows: with probability $\frac{1}{2},$ the register is left unchanged; with probability $\frac{1}{4},$ the two bits of the register are exchanged $(01$ becomes $10,10$ becomes $01,00$ remains $00,$ and $11$ remains $11)$; and with probability $1/4,$ the right bit is flipped $(01$ becomes $00,00$ becomes $01,11$ becomes $10,$ and $10$ becomes $11).$ After the register has been manipulated in this fashion, the left bit is observed.

Suppose that on the first two time steps $(t=1$ and $t=2),$ we observe the sequence $0,0.$ In other terms, the observed bit at time step $t=1$ is $0,$ and the observed bit at time step $t=2$ is also $0.$

$.$ Show how the register can be formulated with a temporal model $($a Hidden Markov Model$)$: What is the probability of transitioning from every state to every other state? What is the probability of observing each output $(0$ or $1)$ in each state?

$2.$ Use the filtering equation $($forward algorithm$)$ to determine the probability of being in each state at time $t$ after observing only the first $t$ bits, for $t=1,2.$

$3.$ Use the smoothing equation to determine the probability of being in each state at time $t=1$ given both observed bits.

$4.$ Use the prediction equation to determine the probability of observing $1$ in the next time step $t=3.$