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The effect of this card is only activated when it is "destroyed". It will not activate if Tributed, discarded, or sent to the Graveyard.
When "Twin-Headed Behemoth" is Special Summoned by its own effect you can activate "Card of Safe Return" and draw 1 card.
When "Twin-Headed Behemoth" is Special Summoned by its own effect and is equipped with "Megamorph" it is the original ATK/DEF that is doubled or halved. (ATK 3000 / DEF 2400 if doubled, ATK 750 / DEF 600 if halved)
The opponent cannot activate "Bottomless Trap Hole" when "Twin-Headed Behemoth" is Special Summoned by its own effect, because its ATK is already changed to 1000 at the time of Summoning.
The condition that this card's effect can only be used once per Duel cannot be reset. So if it is Special Summoned by its own effect and later removed from play, flipped face-down, returned to your hand, destroyed and Special Summoned, or shuffled into your Deck, you STILL cannot use its effect a second time.
If removed from play temporarily by the effect of "Dimensionhole" or "Interdimensional Matter Transporter", it will return with its original ATK/DEF, but its condition is not reset.
If flipped face-down, and then Flip Summoned, it will have its original ATK/DEF, but its condition is not reset.
If equipped to "Relinquished" and destroyed, you can use the effect of this card and it will be Special Summoned on your side of the field with ATK/DEF of 1000 points. (If you have not already used this effect once.)
If this card is controlled by your opponent and is destroyed, it will be Special Summoned on your side of the field with ATK/DEF of 1000 points. (If you have not already used this effect once.)
If "Twin-Headed Behemoth" would be Special Summoned by its own effect, but your Monster Card Zones are full, or the effect of "Scapegoat" prevents you from Summoning, it still considered that its effect was activated, and you cannot use it again in the current Duel.
You may still Special Summon "Twin-Headed Behemoth" with the effect of "Monster Reborn", etc. after its effect has been used once.
If you activate "Monster Reborn" to Special Summon "Twin-Headed Behemoth" in the turn it was destroyed, its effect has not yet been used, and it will activate the next time it is destroyed.