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The problem with using a power supply is the starting current requirement for the motor which may be quite high. I'd be replacing the battery, they're only about $25-$30 for the size you have. I would also be investing in a suitable charger if the only charging option you have is the solar panel. Gel batteries aren't designed to be cycled to any extent, although you can get deep cycle variants. The charging requirements for Gel Batteries are quite specific, particularly if they are run dead flat so using a charger that is specifically designed for gel batteries will help prolong the life.
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