Help with difference between using [verb]ませんか/[verb]ましょう/[verb]ましょうか

Hello! De-rusting my Japanese and going back the he basics. The article I’m going through has these three grammar points back tk back with a very similar example sentence. I’m trying to figure out the nuance between these three forms.

一緒に映画を見ませんか。

映画を見ましょう。

映画を見ましょうか。

It is my understanding that sentence 1 is phrased as an invitation, 2 as a suggestion, and 3 as a suggestion (but polite?). Is that right? Or am I’m missing a piece here.

TIA ❤️

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