Design the logic for a security system.

$1.$ The system contains a door sensor$,$ a window sensor$,$ and a key activation switch.

$2.$ The activation switch arms and disarms the system.

$3.$ The key activation switch outputs a high when it is activated and a low when it is de$-$activated.

$4.$ If the activation switch is on$,$ opening the window will set off the alarm immediately.

$5.$ If the activation switch is on$,$ opening the door will start a $3$ second timer.

$6.$ If the key activation switch is not turned off within the $3$ seconds, the alarm is set off.

$7.$ The door sensor and window sensor output a low when they are opened.

$8.$ The $3$ second timer requires a high to activate and outputs a high when the timer expires.