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Of course an ad network would pay a search engine for clicks (and a premium for exclusivity, perhaps). There's certainly some synergy, but it's also quite possible to run them as separate businesses.

Operating systems might have to get lumped into the 'everything else' bucket if they don't generate enough revenue through charging manufacturers or getting paid by ad networks.

Splitting up these things may not make any of the markets (search, ads, OSes, other consumer services, other enterprise/cloud services) more competitive, but it would prevent using monopoly positions in search and ads to subsidize Google products in other markets.