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In a Two-Headed Giant game, life gained by your teammate won't cause South Wind Avatar's last ability to trigger, even though it caused your team's life total to increase.

If you gain an amount of life "for each" of something or "equal to the number" of something, that life is gained as one event and South Wind Avatar's last ability will trigger only once.

Use the creature's toughness as it last existed on the battlefield to determine how much life you gain with South Wind Avatar's second ability.

Each creature with lifelink dealing combat damage causes a separate life-gaining event. For example, if two creatures you control with lifelink deal combat damage at the same time, South Wind Avatar's last ability will trigger twice. However, if a single creature you control with lifelink deals combat damage to multiple creatures, players, planeswalkers, and/or battles at the same time (perhaps because it has trample or was blocked by more than one creature), the ability will trigger only once.

South Wind Avatar's last ability triggers just once for each life-gaining event, whether it's 1 life from Illegitimate Business or 3 life from Guac & Marshmallow Pizza.