Clarification needed: Sword of Nuln and Heal Damage

Name: Sword of Nuln
Cardtype: Equipment
Clan: Tremere
Cost: 1 pool
Unique melee weapon.
Strike: strength damage. Vampires must spend twice the normal amount of blood to heal the damage from this weapon's strikes.

6.4.6 Damage Resolution
[...] For each point of damage inflicted on a vampire, he must burn one blood to heal the damage. A vampire can burn all of his blood if needed, and doing so does not have any other negative effects on the vampire. If a vampire cannot heal all the damage (i.e., more damage is inflicted than he has blood with which to heal), he burns blood to heal what damage he can, and the unhealed damage leaves him wounded. [...]

A vampire with 3 blood is hit by Sword of Nuln with +1 Strength. No damage prevention cards are played.
Does the vampire have to try to prevent the second point of damage and burn his third point of blood before going to torpor, or will/can he go to torpor with one blood?

Thanks in advance!

The vampire burns all blood, then goes to torpor.

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