Aspirin against vaccine induced blood clots?

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I may have to get vaccinated against my will, along with my wife, so I am thinking about ways to mitigate the risk.

Aspirin is a popular, readily available blood thinner and is being prescribed for Covid.

Would Aspirin be able to prevent vaccine induced blood clots? If it could, why isn't it given to risk groups for blood clots, who are getting vaccinated?

I seem to have read somewhere, that Aspirin wouldn't do anything good in this case, but forget where and can't find anything about it.