The system is resisting your ability to assert control, so 'playing along' and developing momentum, learning the details of the run, then using that momentum to force a change is a valid way to get a toe in the door there.
Another way would be to Form a Suspicion, then Confirm the Suspicion. This lets you mark Verity Exp. or Assertion Exp. as you prefer. So if your explorations lead you to suspect some hidden aspect or Lost society, you can explore that.
You can also gain experience by setting an assertion goal when you Begin a Foreverie Run, and complete that objective. You don't have to use Assertion to get a victory here, obviously it would help.
(I had other material I'd not finished writing up to meet the deadline, including 'spending' from the experience bars to gain access to assets or other benefits. As it stands at the moment you can essentially become Neo by getting Assertion Experience all the way to the top, but that would likely take a whole campaign to achieve so I left it as is for this jam.)