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Currently, headless employs post task to asynchronously dispatch
events.

This, however, breaks the assumption that certain events
should be dispatched before the message response is delivered. This
patch restores the invariant.

R=caseq, alexclarke

Change-Id: Ie58c0ee060df7874a6e378ff6962a82c303f864b
Reviewed-on: https://chromium-review.googlesource.com/1048725Reviewed-by: Alex Clarke <alexclarke@chromium.org>
Reviewed-by: Andrey Kosyakov <caseq@chromium.org>
Commit-Queue: Andrey Lushnikov <lushnikov@chromium.org>
Cr-Commit-Position: refs/heads/master@{#556783}