Implement a centralized logging module to replace print()

[Arseni10Lk]

The codebase currently relies on standard print() statements for output. This makes it impossible to filter output by severity or automatically save logs.

Task:

• Implement a centralized logger using Python's built-in logging module.
• Replace all print() calls in IADE_Core/ with the appropriate logger calls.
• Ensure the logger is configured to also record/dump actual Python objects and JSON results when a solver finishes, not just text strings.