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How to set up the update background task in order the Flash updates as soon as possible
upon first run of operating system after the appropriate update was released by Adobe?
Are you distributing Flash Player within your organization? If so you can license Flash Player for distribution and then host the Background Update resources locally. This would require you to post the new resources to your local sever whenever there is a new release. You can find more information on this, including information on licensing Flash Player and the requirements to setting up a local server in the Flash Player Administrator's Guide.