Do you speak English or French?—I don't speak ____ of them.B.both C.neither

Do you speak English or French?—I don't speak ____ of them.B.both C.neither
—Do you speak English or French?—I don’t speak ____ of them.A.none B.both C.neither D.either
老师说选C,我觉得应该选B.
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网上找到个解释：
“Do you like any coffee or tea?
I don't like ____.I prefer water.
A.both B.either
如果两者否定,应该用either
both用于否定句是指部分否定,即喜欢其一,这里显然是两者都不喜欢”
我就知道为什么both不对了,但即使这样也不应该是选C啊?