Flutter PostHog Analytics Integration

I have an existing Flutter project that needs PostHog wired in from start to finish. Once the SDK is added I want to see three kinds of data flowing to my PostHog workspace: user interactions, page views, and a handful of custom events that I will specify in comments inside the code.

What I need from you is straightforward: update the project, initialise PostHog on app launch, add the event-tracking hooks in the right screens and actions, then return the modified codebase so I can pull it, run it, and immediately watch those events appear in PostHog.

Please include any new dependencies in pubspec.yaml, keep the code clean and well commented, and verify that at least one sample event of each type shows up in the PostHog dashboard before you hand the project back.