Hello,
This is only a super simple addon that pre-fills the input and most things you see is doing ChatGPT page :).
But actually this is really terrible, it will submit the query too quickly and that will trigger some crazy bug in their page causing two or more history items to be created...
I'll try to fix this in upcoming days!

The main feature I use is the "Alt + i" shortcut that opens it (in my case in a new window).

Regarding the integration with the search results page, this is sadly not possible since my addon doesn't have permission to alter other pages.
Also, it can't work in background, that would require using their official API which is not free.

I'm sorry for the bad news :(

Best Regards,
Juraj Mäsiar