Refactor client connection management and improve code formatting

[NanaMein]

Overview

This PR introduces clean architectural decoupling for connection management by migrating application startup/shutdown logic into a dedicated lifespan module and utilizing a centralized global application state (app_state).

Key Changes

• Reduced main.py Verbosity: Extracted FastAPI lifespan context manager logic into a new dedicated module (app/core/lifespan.py).
• Centralized Application State (app_state): Introduced a global state container to manage heavy client instances (Database/PGVectorStore, Groq LLM client, etc.).
• Decoupled from Request Context: Eliminated the need for services and classes to depend on or extract connections from the FastAPI Request object.
• Eager Initialization: Ensured all external connections and sockets are fully initialized during server startup before the application accepts any incoming HTTP traffic.

Benefits

• Better Performance & Cleanliness: Avoids lazy-loading overhead or request-scoped connection passing.
• Maintainability: Clear separation of concerns between HTTP routing (main.py), startup lifecycle (lifespan.py), and service logic.

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