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Have basic electricy stuff
Pull the start capacitor and check for continuity, they often fail short circuit. It's not definitive if it looks open circuit, but a short means it's definitely fucked. Add an LCR meter to your armoury, a cheap one off of ebay for ~£30 will be sufficient for this kind of go/no go test to see whether the capacitor is behaving as a capacitor of the right value ±10%
What tools do have at hand? Do you have the circuit diagram?