Normalize and validate an email address

Write a function normalize_email(email) that takes an email string and returns a normalized email string when the input is valid, or None when invalid. Normalization rules: - Strip leading/trailing whitespace and lowercase the whole address. - If the domain is gmail.com or googlemail.com, remove dots in the local part and remove any plus-tag (text starting with '+') before the '@'. For other domains, leave the local part (except trimming and lowercasing). Validation rules (apply after trimming and lowercasing): - There must be exactly one '@'. - Local part must be non-empty and cannot start or end with '.' - Domain must contain at least one '.' separating labels; labels must start and end with an alphanumeric character (letters or digits). - The email must not contain spaces. Return the normalized email string if valid, otherwise return None.

Implement a function that defensively normalizes emails (strip, lowercase, optional Gmail-specific cleanup) and validates format.

- Do not use external libraries. Keep checks simple and defensive. - Emails may contain unusual inputs; function must not raise exceptions for malformed inputs.