WILL PRINT A PAGE THEN SAYS CAN'T AUTHENTICATE CARTRIDGE AND WON'T PRINT

SOURCE: Lexmark S305 won't authenticate ink non brand

Lexmark does not recommend the use of refilled, re-manufactured or clone cartridges. You might not get the print quality you expect from ink that is specially formulated by Lexmark. Also, when you use off-brand inks and cartridges, there is no way for Lexmark to be sure that materials and engineering choices used by a third party will not result in poor print quality or damage to your printer.
I highly recommend that you return your third-party cartridge if you still can get a refund and purchase genuine Lexmark cartridges instead.
Here's a tip to save on ink: A 100 "standard yield" cartridge for your model has a low sales price, but contains less ink. Your best value is a 100XL "high yield" cartridge. These have a higher sticker price, but contain almost 3 times as much ink as their standard yield cousins.

For the most optimal output experience with your model, I recommends that you update your printer firmware. The firmware helps ensure that you receive the highest yields of ink available from the cartridges, which offers the greatest value and means fewer necessary cartridge replacements. Please visit the Lexmark website to download and install the Lexmark S300 Series Firmware Update. If more than one download is listed for your model, choose the first download in the list. If it is the wrong firmware level, an error message will tell you which firmware level is correct.