Cache package databases

In theory these files could be cached, to speed up updating:

$ ls /var/lib/pacman/sync/
clang32.db         clang64.files         mingw32.db         mingw64.files      staging.db
clang32.db.sig     clang64.files.sig     mingw32.db.sig     mingw64.files.sig  staging.files
clang32.files      clangarm64.db         mingw32.files      msys.db            ucrt64.db
clang32.files.sig  clangarm64.db.sig     mingw32.files.sig  msys.db.sig        ucrt64.db.sig
clang64.db         clangarm64.files      mingw64.db         msys.files         ucrt64.files
clang64.db.sig     clangarm64.files.sig  mingw64.db.sig     msys.files.sig     ucrt64.files.sig

It will still hit the repo and mirror but should result in a 304 in most cases, assuming the mtime gets restored correctly.

Question is if we should cache them separately. Probably better, assuming we restore in parallel, since they might change more often than packages and invalidate the caches needlessly otherwise.

Start by tracing how this GitHub Action updates MSYS2 and handles the package database path shown in /var/lib/pacman/sync/. Determine whether those databases should be cached separately, then verify that restoring their mtimes produces conditional 304 responses without unnecessarily invalidating package caches.