Morning Aleksei
Welcome to the Forum
Try You tube for removal Videos they normally have something. To be sure you don't break or damage any of the trim parts get a set of Proper Trim removal tools they are not expensive and will prevent damage to trim and fixings.
The CIC fault could be a loose connector or a failing joint on a circuit board only stripping and testing will tell. Once out you can make the decision to repair or replace it.
Dave