No printing. The cartridge light blinks and the printer won't print. I've tried removing and

Question

The cartridge light blinks and the printer won't print. I've tried removing and reinserting the cartridges and it doesn't make a difference, even letting the levers push the cartridges into place instead of pushing them down by hand. I'm looking for ideas of what specifically to check for an clean and an reluctant to buy new cartridges to test with because I can't use them in another printer if this one isn't going to work.