Actions don't have access to the document name (where the file name is). Scripts do, however.
If your products are sequentially numbered, you're in luck. You can overide the "Save as command" and make sure that your naming uses a sequence which you can set to the same starting number as your product. Here is an example of how to configure File -> Automate -> Batch (assume the set and action are your hand built one)
Instead of "Document name" you'd type "PRODUCT_" per your example and choose a 3 digit Serial number.
The "Starting Serial #" is where you can change if your products are numbered say, 101 and up.