synchronization for price and stock

Hello!
We have an issue with products price update.
We understood that the stock is not updated with your plugin, but we need to see which items are already unavailable on amazon and delete them from the website.
For that we could use price update – so we could spot the products with no price and remove them.
But now the synchronization doesn’t work.
could you please help with these issues:
1. How to make price synchronization work
2. How to spot out of stock products in time?

thank you!
Lena