Type Drive API for the Prompt Gateway code

[junr03]

The prompt gateway code is difficult to follow given all the transitions it can take. Evidence of this is the crate/prompt_gateway/context.rs docs (and the fact that this file has to live on its own).

This code is ripe to be re-architected. Some inspiration:
https://willcrichton.net/rust-api-type-patterns/state_machines.html
https://hoverbear.org/blog/rust-state-machine-pattern/