You return 1 card from your hand to the Deck after drawing 1 card from your Deck. The last thing to happen is not you adding a card to your hand, so your opponent cannot activate "  Drastic Drop Off".
After returning 1 card from your Hand to the Deck with this effect, shuffle your Deck.
You can activate this effect even if you have no cards in your hand.
The player who activated the effect of " Number 56: Gold Rat" chooses which card to return to the Deck. He can even choose to return the same card that he drew.
You can activate the effect if there is one card left in your Deck. You cannot activate the effect if there are no cards left in your Deck.
If " Number 56: Gold Rat" is not face-up on the field when resolving its effect, then you still draw a card and then return a card to your Deck.
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Konami OCG FAQ: Xyz Monster > Number 56: Gold Rat
Konami OCG FAQ: Are drawing and returning to your Deck for "Number 56: Gold Rat" done simultaneously?
Konami OCG FAQ: Can you activate the effect of "Number 56: Gold Rat" if there is a "Protector of the Sanctuary"?
Konami OCG FAQ: Does the player choose which card is returned to the Deck for the effect of "Number 56: Gold Rat"?
Konami OCG FAQ: Can you activate the effect of "Number 56: Gold Rat" if you have one or fewer cards left in your Deck? ↑
Konami OCG FAQ: If "Number 56: Gold Rat" is not on the field at resolution, then do you still do the effects which have you draw and discard from your hand?
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