A designer gives you the following state machine of a proposed system:

input: temperature :$R$

outputs: heatOn, heatOff : pure

tam$$anations hant $$ n

However the designer proposes the following changes:

there is only one temperature threshold, and to avoid chattering the thermostat simply leaves the

heat on or off for at least a fixed amount of time. In the initial state, if the temperature is less than or

equal to $20$ degrees Celsius, it turns the heater on$,$ and leaves it on for at least $30$ seconds. After

that, if the temperature is greater than $20$ degrees, it turns the heater off and leaves it off for at least

$2$ minutes. It turns it on again only if the temperature is less than or equal to $20$ degrees.

$($a$)$ Design an FSM that behaves as described, assuming it reacts exactly once every $30$ seconds.

$($b$)$ How many possible states does your thermostat have? Is this the smallest number of states

possible?