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There are two ways this can go depending on R4C34:
Red must pass those two cells, because it's coming from box 3 and can't return to box 3 to get around yellow.
Green must pass those two cells, because it's already going from box 6 to box 5 and can't return to box 6 to get around yellow.
Therefore, of R3C34, one is red and the other is green.
You just proved that R3C3 can't be red.