Is it important to you to have a key? A fair amount of 20th century music tried to scrupulously avoid a "key". There are many songs that don't start with the chord of the key, but most will end on the key. It's where it feels finished. The G7 to C progression at the end suggests a tonal center of C, but some of the chords don't imply a typical harmonic analysis. I wouldn't worry about it.
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