static/shared/header: perform search in currently selected mode
static/shared/header: perform search in currently selected mode

Added a hidden input element to pass the current search
mode to additional queries made from the search results
page.

Change-Id: Ia9dc1bb209d85a8e8ad1ba531b12ccdb64037ab6
Reviewed-on: https://go-review.googlesource.com/c/pkgsite/+/347554
Trust: Jamal Carvalho <e0195770807aa8c82b0b128d9c0423b5ad035172@golang.org>
Run-TryBot: Jamal Carvalho <e0195770807aa8c82b0b128d9c0423b5ad035172@golang.org>
TryBot-Result: kokoro <2ac7b1f3fa578934c95181d4272be0d3bca00121@google.com>
Reviewed-by: Julie Qiu <8d32267b6b4884cf35adeaccde2b6857ae11aace@golang.org>

golang/pkgsite,golang/pkgsite,golang/pkgsite,golang/pkgsite