feat: implement real-time study group rooms (closes #502)

[singhanurag0317-bit]

Summary

Resolves #502

This PR implements the requested Study Group Rooms feature allowing users to join collaborative focus sessions in real-time.

Changes

• Backend (server.js, socket.js):
  • Integrated socket.io for real-time WebSocket communication.
  • Added in-memory room tracking to handle participant limits, host assignment, and synchronized timer states (duration, remaining, isRunning).
  • Designed socket events to handle goal updates, timer starts/pauses/resets, and participant join/leave broadcasts.
• Frontend (index.html, room.js):
  • Added a "Study Rooms" tab to the main sidebar.
  • Built a dynamic lobby interface to either create a new room or join via a generated invite code.
  • Implemented the active room dashboard, featuring a synchronized circular SVG timer, shared goal input, live invite link display, and a real-time participant list.
  • Restricted timer controls (Start/Pause/Reset) to the room Host.

Verification

• Verified that multiple clients can join the same room via an invite link.
• Verified that the participant list dynamically updates as users connect and disconnect.
• Verified that starting, pausing, or resetting the timer immediately synchronizes across all connected clients.
• Verified that only the room host is allowed to modify the timer.