fix : duplicate routing logic and multiple module import

[shivani406]

# Pull Request

## Description
This PR solves critical initialization issues in the authentication router file (app/api/routes/auth.py) that caused application crash during runtime.

Changes made :

Fixed the NameError by ensuring router = APIRouter(prefix="/auth", tags=["auth"]) is declared at the top of the file before any endpoints use it and eliminated the duplicate APIRouter().

Cleaned up redundant, duplicated import statements at the top and middle of the file to improve code design and readability.

## Type of Change
- [ ✔️] Bug Fix
- [ ] New Feature
- [ ] Documentation
- [ ] UI Improvement
- [ ] Refactor

## Related Issue
Closes #81 

## Checklist
- [✔️ ] Code tested locally
- [ ✔️] No unnecessary files added
- [ ✔️] Documentation updated

[vercel]

@shivani406 is attempting to deploy a commit to the devloperdevesh's projects Team on Vercel.

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[Karanjot786]

Hey @shivani406! Saw your work on GSSoC 2026.

We are building TermUI, a TypeScript terminal UI framework with React-style hooks and JSX, rendered entirely in the terminal.

We have 99 unassigned GSSoC issues open, including well-scoped bug fixes with clear repro steps. Your TypeScript background transfers directly.

• Open issues (GSSoC 2026)
• TermUI on GSSoC

Karanjot, TermUI maintainer

[devloperdevesh]

Hi @shivani406,

Thanks for fixing the duplicate auth router issue.

This PR currently has a merge conflict with the latest main branch in app/api/routes/auth.py.

Please update your branch with the latest main branch, resolve the conflict, and push the changes.

Once conflicts are resolved, I will review and merge.

Thanks!