$(50$ pts$.)$ A machine's status is checked at the end of every hour and can be classified in one of three ways: $1$ operating properly, $2-$operating in a degraded condition, $3-$ failed. The machine's status at the end of an hour is only dependent on the status of the machine at the beginning of the hour. Thus, if we let $x(t)$ denote the status of the machine or $3$