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This is part of the work of migrating the existing functionality to use
the new infra components.

We factor the task management out of the context manager implementation
and into a transition called the Startup. This includes metrics
reporting as well, which make more sense in the context of the startup
object.

Bug: b/175360169
Change-Id: I49cc7e7740e76eda873ee7104924b3e344348154
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2584631Reviewed-by: Daniel Ng <danielng@google.com>
Commit-Queue: Nic Hollingum <hollingum@google.com>
Cr-Commit-Position: refs/heads/master@{#836756}