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The burning sensation you felt is a sign of irritant contact dermatitis
and good that you washed it off in 3-5 minutes, otherwise the damage would have been more. The skin near your vagina, is very sensitive and it is better to avoid depilatory creams over that skin.
Irritant contact dermatitis should be treated with topical steroid
cream and a combination of that with an antibiotic is ideal an apply it twice a day. You can keep cold
compresses if it is burning. In few days time it will become dry and flaky and fall off. Do not pick on the flakes. You may apply a moisturising lotion three times a day.
Hope this helped