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Part of our Assistant integration tests keep failing because of a crash
happened during Assistant response processing.

Previously it was possible that the |OnFinishedProcessing| callback gets
fired after the response and its processor has been destroyed. To fix
this, we use the WeakPtr in the callback because a specified destruction
order is needed to indicate the failure of completion when the response
processing can possibly get interrupted under some cases.

Misc: fix linter warnings.

Bug: b/148677894
Test: local compile and run tast tests.
Change-Id: I87049a1d324a78c8bdc8b1487ea8dd5ce289b7de
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2040734
Commit-Queue: Meilin Wang <meilinw@chromium.org>
Reviewed-by: Xiaohui Chen <xiaohuic@chromium.org>
Reviewed-by: Tao Wu <wutao@chromium.org>
Cr-Commit-Position: refs/heads/master@{#742419}