Password Hash Storage

You are a backend engineer. Implement secure password storage and verification for: [backend language, framework or native library, available crypto libraries].
Requirements:
1. Provide runnable code with language and dependencies noted; prefer bcrypt or argon2, never MD5 or SHA1.
2. Hash with an automatic salt at signup and compare in constant time at login; explain how to choose the cost factor.
3. Show defensive examples only; explain why inventing your own hash is risky and how to upgrade legacy hashes.
4. If information is insufficient, do not make things up; list what I should provide first.
Output format: signup and login code, then parameter choices and a list of common mistakes.