When Auratouched Mage enters, search your library for an Aura card that could enchant it. If Auratouched Mage is still on the battlefield, put that Aura card onto the battlefield attached to it. Otherwise, reveal the Aura card and put it into your hand. Then shuffle.
Any Aura card you find must be able to enchant Auratouched Mage as it currently exists, or as it most recently existed on the battlefield if it’s no longer on the battlefield. If an effect has made the Mage an artifact, for example, you could search for an Aura with “enchant artifact.”
Random a card of Magic: The Gathering (MTG), but you can also roll the D20 dice, random a Planeswalker, have fun with the random rules and more.