If I have put Control Magic on an opponent's creature, and then choose that creature as one of my targets for Puca's Mischief, what happens?

701.10b When control of two permanents is exchanged, if those permanents are controlled by
different players, each of those players simultaneously gains control of the permanent that was
controlled by the other player. If, on the other hand, those permanents are controlled by the
same player, the exchange effect does nothing.

613.1. The values of an object’s characteristics are determined by starting with the actual object. For a
card, that means the values of the characteristics printed on that card. For a token or a copy of a
spell or card, that means the values of the characteristics defined by the effect that created it. Then
all applicable continuous effects are applied in a series of layers in the following order:

613.1b Layer 2: Control-changing effects are applied.

613.6. Within a layer or sublayer, determining which order effects are applied in is usually done using a
timestamp system. An effect with an earlier timestamp is applied before an effect with a later
timestamp.

613.6b A continuous effect generated by the resolution of a spell or ability receives a timestamp at
the time it’s created.

613.6d An Aura, Equipment, or Fortification receives a new timestamp at the time it becomes
attached to an object or player.

Can I exchange control of the creature?

Do I get it back again?

611.2a A continuous effect generated by the resolution of a spell or ability lasts as long as stated by
the spell or ability creating it (such as “until end of turn”). If no duration is stated, it lasts until
the end of the game.

Follow up question, then. What would happen if I someone else control of the Aura? Would they get the creature? And this goes both before and after I've traded away the creature.

109.5. The words “you” and “your” on an object refer to the object’s controller, its would-be controller
(if a player is attempting to play, cast, or activate it), or its owner (if it has no controller). For a
static ability, this is the current controller of the object it’s on. For an activated ability, this is the
player who activated the ability. For a triggered ability, this is the controller of the object when the
ability triggered, unless it’s a delayed triggered ability. To determine the controller of a delayed
triggered ability, see rules 603.7d–f.

613.6. Within a layer or sublayer, determining which order effects are applied in is usually done using a
timestamp system. An effect with an earlier timestamp is applied before an effect with a later
timestamp.
613.6a A continuous effect generated by a static ability has the same timestamp as the object the
static ability is on, or the timestamp of the effect that created the ability, whichever is later.
613.6b A continuous effect generated by the resolution of a spell or ability receives a timestamp at
the time it’s created.
613.6c An object receives a timestamp at the time it enters a zone.
613.6d An Aura, Equipment, or Fortification receives a new timestamp at the time it becomes
attached to an object or player.
613.6e A permanent receives a new timestamp at the time it turns face up or face down.
613.6f A double-faced permanent receives a new timestamp at the time it transforms.
613.6g A face-up plane card, phenomenon card, or scheme card receives a timestamp at the time
it’s turned face up.
613.6h A face-up vanguard card receives a timestamp at the beginning of the game.
613.6i A conspiracy card receives a timestamp at the beginning of the game. If it’s face down, it
receives a new timestamp at the time it turns face up.
613.6j If two or more objects would receive a timestamp simultaneously, such as by entering a zone
simultaneously or becoming attached simultaneously, the active player determines their relative
timestamp order at that time.

So if a player steals a creature with Control Magic then returns it through Puca's Mischief, the Aura will remain in their control while the creature is controlled by their opponent. Does this mean the player can later exchange control of Aura and enchanted creature, effectively getting the creature back without giving anything useful back?