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I am looking to build a circuit that would control an output relay. This would be done in 12V and the sequence will be initiated by a manual switch. I will need an adjustable time delay (possibly displayed time) after the switch is released, then the output would be random before shutting off. The sequence would not restart until the button was pressed and released again. The time after the button release would be from 0 milliseconds to 3 seconds. The "on" time for the output to turn on the relay would be from 500 milliseconds to 1 seconds

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Especially if you actually need a display of sorts, but even without, this sounds easiest when implemented in a microcontroller.

You don't really need anything fancy - watch for your button input to go high (or low, depending on how you wired the button). Wait. Toggle output pin. Wait. Toggle again.

On that output pin, you'll need some transistor and a flyback diode to switch the 12V for your relay. There's inexpensive N-channel MOSFETs that integrate the diode and can directly be driven with logic levels.

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