"seem to ~" takes a verb infinitive. Plain infinitive would be "to avoid"; "to be avoiding" is the continuous infinitive: Everyone avoids Nick. -> Everyone seems to avoid Nick.
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"seem to ~" takes a verb infinitive. Plain infinitive would be "to avoid"; "to be avoiding" is the continuous infinitive:
Everyone avoids Nick. -> Everyone seems to avoid Nick.
Everyone is avoiding Nick. -> Everyone seems to be avoiding Nick.