If you take action to automatically end persistent damage, do you still take the damage that turn?

I’m preparing to run a Pathfinder 2e game with an alchemist player and trying to fully understand persistent damage to prepare.

When you take persistent damage, you don’t apply it until the end of your turn. After taking that damage, you make a DC 15 flat check to see if you remove the condition. But you or your allies can take actions to aid in recovery from the condition. For example, if you are on fire and you dump water on yourself, it looks like you still take the full amount of fire damage but then your flat check to remove the condition is reduced to DC 10.

There is also a way to automatically end the condition (quote from CRB under persistent damage condition):

Automatically end the condition due to the type of help, such as healing that restores you to your maximum HP to end persistent bleed damage, or submerging yourself in a lake to end persistent fire damage.

In this case, you don’t make a flat check to see if it ends, it just automatically ends. But this check that you automatically succeed on would have occurred after you already took the damage.

So do you still take the damage and then the check to remove the condition is automatically successful? Or because the condition was removed do you ignore the damage that turn?