I'm gonna just splice together the wires with a butt crimp and wrap them in electrical tape and call it good.
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At the minimum, use these.
www.amazon.com/gp/product/B07L29DLGN/
Yes, you can use a multimeter instead of a test light, but a test light is often faster for simple troubleshooting because you don't have to take your eyes off your work.
I meant to test across the plug, not the fan.