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Sounds like you just need to adjust your rear brake.  I just did the same thing the other day.  Look at your rear brake drum where the brake cable connects to the lever that activates the brake when pulled.  At the end of that cable is a nut.  Turn that nut clockwise once or twice and check your brake lever.  The nut has a saddle so it turns in 180 degree increments.  Keep doing that until you get the brake lever to engage at the point that you're comfortable with.  I set mine to engage about 30% pull.