Your list of contacts in the People App, should be linked to your Microsoft account and therefore preserved through any cleaning and re-installation.
You can test and confirm this by simply using a computer from any source (library, cyber-cafe, friend, neighbor, relative, etc.), Open a web browser, go to https://account.microsoft.com/account
Login and check to confirm that the contents of your People App (online) is as you would expect it to be.
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