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urgeUrge incontinence you have not to treat yourself unless you have a medical diagnosis for your problem. To do otherwise is likely to be unsuccessful and to confuse things.
Your risk of syphilis from the exposure you describe is low. Syphilis is a rare disease. If you have a genital sore which has been present for more than two or three days, a syphilis blood test will be negative. If you are in this situation and have a negative blood test, you do not need to worry about syphilis. If the sore is also present you could also be tested with a herpes culture- these are the two main causes of genital ulcers (sores).
Doxycycline can cure syphilis but the recommended dose is 100 mg twice daily for 14, not 10 days. This would also cure chlamydia.
Once more, my advice is to go to a health care provider and seek testing so that you can find out what you have, if anything, and whether you need treatment. Please do not try to self medicate. Thanks. EWH