The #1 piece of advice I can offer is to update your Maxon App. Maxon app v2023.1.2 was released in December 2022. With this update, they fixed an issue where Red Giant serial numbers were failing to Red Giant license plugins.
If you bought Trapcode Particular v4 and recently reinstalled it from the installer in the original email, you may find that you are no longer able to activate your serial number. What has happened is that you inadvertently installed v5. No worries. Here is how to fix it. 1e1e36bf2d