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Almost of all of the code in this class is shared acoss all instances of
the tempate. This introduces a ThreadSafeForwarderBase class and moves
the logic into it.

No functional changes.

Bug: 1016022
Change-Id: I515c472f8a6fdc97fdd25d90db596d23a89ce543
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1871195Reviewed-by: Oksana Zhuravlova <oksamyt@chromium.org>
Commit-Queue: Ken Rockot <rockot@google.com>
Cr-Commit-Position: refs/heads/master@{#708012}