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You snake the drain through the overflow not the drain in the bottom of the tub. You remove the face plate and pull out the linkage then snake through there.
And a power snake is the way to go. Hand snakes seldom do the job. You can rent one at most tool rental outlets.
I don't understand your last comment. Are you saying the toilet is backing up too? into the tub? if so then you have a main line stoppage and you'll have to snake the main with a motorized snake that's considerable bigger than the sink snake.