Lip allergy? Worry

My lips crack when I wear lipstick.

I think that is especially true the stronger the red color.

After applying the lipstick, about 30 minutes later, the area around my lips started to smudge and something like pus would come up.

If my skin gets worse after applying lip balm when going out, it takes more than a week to recover.

Seeing that there are lipsticks that don't do that, it seems like it might be due to a specific ingredient in the lipstick....

I'm sure there's no one else like me.