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I'd tend to use "firstly, secondly..." more often than "first, second...", but I have no objection to either, with a couple of caveats.

Should my enumeration end with "lastly" or "finally", I'd...

In my experience in BrE wrt ~ed/ ~t past tenses, some verbs have both variants and are equally common, some have both but one form is more common than the other, and some are only ever used with one....