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I had the same problem. The bleeding is caused by an infection, if not treated early enough the infection will dig deeper into the area between the teeth and gum to the point where flossing, brushes and mouthwash literally do not reach the infected area buried deep between tooth and gum. This is why it continues to bleed even though you feel you are doing the right things.
The solution in my case was to have an operation done on my gums whereby they literally cut off some of the top of the gum, peel it back, clean the infection out then let the gum "stick" again to the tooth. It was not a pleasant operation, its usually done by a specialist (which is why your dentist has probably not said anything: he needs to refer you!) under general aneastheic. These days, though, I have heard that it can be treated through lasers.
Hope this is ok. The reality isif you do nottreat this it will only get worse over the years to the point that eventually the gums will completely detach from the teeth with obvious results!
Good luck, hope you keep a beautiful smile! :)