Is the correct phrase game the system or gain the system?

It's GAME the system. It meansto manipulate and take unfair advantage of loopholes in rules and regulations in order to make that system of rules and regulations work to your own advantage in risky, typically illegal, schemes.

Synonym to "play fast and loose" (=game) "with the rules" (=system)

Example: He came to this country illegally on the extectation that he could get away with it, take advantage of the benefits, and game the system.