If you really, really need to get out fast and aren't picky about where you land so long as it isn't here, consider Svalbard. Living there is quite difficult, but _getting_ there is incredibly straightforward.
If you have significant money or specialized skills you have more options, and there are a bunch of countries that actively try to make it easy to immigrate.
There are also a variety of south and central American countries, but I'm less familiar with those options.
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