[idahime]

雨が降って人は南口に出たり入ったりしました。

I need this sentence to say:
It was raining and people were coming out and entering the southern exit by turns. (in polite Japanese)
I don't know how to manage it since I feel like the te-form of the first verb needs imashita (iru) at the end of the sentence, but I've been taught to combine tari-tari-forms with a final suru/shimasu/shimashita.

I would really appreciate some advice on how to handle this.
Thanks!