A $2-$bit computer register has four states: $00,01,10$ and $11.$ The initially state is set to $00.$ At each step, the register works as follows:

With the probability $1/2,$ the register keeps unchanged;

With the probability $1/4,$ the two bits switch $($e$.$g$.,01->10)$;

With the probability $1/4,$ the second bit flips $($e$.$g$.,01->00$ or $00->01)$

After the change at each step, the value of the first bit is emitted.

Describe this problem as a Hidden Markov Model $($HMM$).$ Draw states and show transitions $($labeled with probabilities$)$ between the states. $($You can draw a picture and upload as a PDF file$)$

What is the probability that the HMM is in a given state at step t $?$

Suppose that, during the first three steps, we observe $0,0,1.$ Use the Viterbi algorithm to find the most probable path of state transitions through the HMM.