How to modify the HTTP request headers using Greasemonkey before a new page is loaded?

I know about the GM_xmlhttpRequest api, but it's not what I'm asking about.

For instance, I want to visit Stack Overflow with a new UA(user-agent), can I do it with a Greasemonkey script?

Solution

Greasemonkey and Tampermonkey only fire after the initial page request is made.
They cannot alter the user-agent.

Modern browsers let you change the reported user-agent in the settings. In addition, there are a plethora of extensions that assist in changing the UA.
(Or you can write your own extension.)

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