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This patch is an alternative approach to only attempting to push gtest
deps once. (The first attempt to do this, landed in
https://codereview.chromium.org/560133002, caused issues and had to be
reverted.)

BUG=400440

Review URL: https://codereview.chromium.org/651923002

Cr-Commit-Position: refs/heads/master@{#299314}