feat: add system prompt to chat tool for better collaborative thinking

- Add CHAT_PROMPT to establish Gemini as senior developer partner
- Update handle_chat to use system prompt for consistent context
- Emphasize edge case exploration and deep thinking in prompts
- Enable thoughtful brainstorming and validation capabilities

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <noreply@anthropic.com>

prompts/__init__.py

@@ -2,7 +2,7 @@
 System prompts for Gemini tools
 """
 
-from .tool_prompts import (ANALYZE_PROMPT, DEBUG_ISSUE_PROMPT,
+from .tool_prompts import (ANALYZE_PROMPT, CHAT_PROMPT, DEBUG_ISSUE_PROMPT,
                            REVIEW_CODE_PROMPT, THINK_DEEPER_PROMPT)
 
 __all__ = [
@@ -10,4 +10,5 @@ __all__ = [
     "REVIEW_CODE_PROMPT",
     "DEBUG_ISSUE_PROMPT",
     "ANALYZE_PROMPT",
+    "CHAT_PROMPT",
 ]