Add query param new chat flow

[Chel-Tbl]

Closes #232.

Summary

• Add /new as an authenticated new-chat entry point with sign-in redirect support.
• Support q on both /new?q=... and /?q=... so external "Open in Toolkit" links can prefill a prompt.
• Auto-submit the prompt when q is present, create the chat through the existing chat API, then replace the URL with the generated chat id.
• Preserve the original destination through /login?redirect=..., including when a signed-in user lands on the login page directly.
• Enable native web search from the initial render when the selected model supports web search.

Verification

• node .\node_modules\prettier\bin\prettier.cjs --check 'src/app/(general)/page.tsx' 'src/app/(general)/new/page.tsx' 'src/app/(general)/login/page.tsx' 'src/app/(general)/_components/chat/index.tsx' 'src/app/(general)/_contexts/chat-context.tsx'
• .\node_modules\.bin\tsc.cmd --noEmit --pretty false
• SKIP_ENV_VALIDATION=1 node .\node_modules\next\dist\bin\next build
• Browser check against http://127.0.0.1:3032/new?q=trace%20audit and http://127.0.0.1:3032/?q=trace%20audit confirmed unauthenticated users are redirected to /login with the original query preserved in redirect.