What key you are in, depends on where you feel the home note ("tonic") is, and that depends on how you play the melody. The chords Am7 - Gm7 - C7 - F could just as well be in a passage that's "in C", or one that's "in F", or many other things. Where's your home note, what note feels like being a natural ending?
Here are two examples of how to use the same chord progression with different melodies.
In the first part we're thinking "in C", and the Bb note (instead of B that's normally in C major) is only used in the "Gm7 - C7" part of the chord progression, so it's kind of like using the F major scale. But after we get to the F then we move back to C major scale and move the melody line towards a C major ending. (And then we actually play a C major as an ending chord)
In the second part we're thinking "in F", and we lead the melody towards an F major ending, and don't really flirt too much with any "what if this is in C major or A minor after all" ideas.
The scale is not the only component in play, how you play the melody line is important too. Do you play like a song that's in C? Or like a song that's in F? Or like a song that's in Am? It's not just the set of notes, it's what you do with it.
Edit: the first example doesn't feel to be very strongly in Am. I added more Am context to the beginning, trying to establish the key there.
In this context, in my opinion, the "Gm7 - C7" bar doesn't move the tonic to F at all, it feels like merely a fancy C7.