[Task]: Drop Continue.dev support and remove from the code.

[lukehinds]

Description

Several times we have tried to figure out workarounds for continue.dev support, but then code changes and we are left trying to fix. Attempts have been made to contribute upstream and put a User-Agent in place, but unfortunately the continue team who must have their reasons, are not willing to provide a UA.

We therefore believe its in the best interests to drop support, as we have no way of knowing how to handle continue requests correctly.

Additional Context

PRs and issues:

continuedev/continue#2988
continuedev/continue#2622

PR made by @jhrozek continuedev/continue#3874