fix(frontend): let users enter password-reset codes

[nourshoreibah]

Summary

• Cognito emails a verification code (CONFIRM_WITH_CODE), not a reset link. The profile "Send Reset Email" button and the forgot-password page claimed a link was sent and left no place to type the code.
• Profile now opens a dialog after the email is sent so a signed-in user can enter the code and a new password (they cannot use /reset-password; AuthGate bounces authenticated users off public routes).
• Forgot-password now collects the code and new password on that page instead of stopping at "Reset Link Sent."

Test plan

• On Profile, click Send Reset Email, enter the emailed code and a valid new password, and confirm the password changes.
• On Profile, submit a wrong code and confirm the dialog stays open with an error (password unchanged).
• From Login → Forgot password, request a code, enter it with a new password, and sign in with the new password.
• Confirm the emailed message still matches the dialog (code, not a link).

Made with Cursor

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🌿 Preview environment — ready ✅

Open: https://d34ajqx96zo93a.cloudfront.net/pr-394/
API: https://am3ulra065.execute-api.us-east-2.amazonaws.com/prod

Shared RDS + Cognito (prod data); DB migrations are not applied here — if this PR adds a migration, endpoints using the new columns will fail until it merges. New commits update this environment in place — a note is posted here on each update. Remove the test-environment label or close the PR to tear it down.

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🌿 Test environment updated in place ✅ — Click here to open. _{updated for 039d82d · logs}