How can I fill value into Search Center default.aspx textbox?

I am using SharePoint 2013 on-premise. By some reason, I need to pass the querystring value into the search center default.aspx. On the page there is only 1 textbox. I can capture the querystring value with Jquery. However I cannot find a way to fill the value into the textbox.

In the F12 developer tool, I found the textbox is quite complex. There is some Javascript running inside the textbox already. Is there any easy way (by adding javascript line into that page) to achieve my target? Thanks.