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Below is the sentence divided into four parts, indicated as (a), (b), (c), (d):

(a) Another baffling change (b)that I notice in him (c) nowadays is that he (d) avoids to speak to me.

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"Avoid" takes gerund phrases as objects, not to-infinitive phrases, so (d) should be "avoids speaking to me." There are no rules that I know of that determine whether verbs take gerunds or infinitives as objects, unfortunately—you just have to learn each case.

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