There are two words. Rein. And reign. One refers to those straps that make a horse go where you want it to. The other refers to the rule of a monarch. Both can be used as either nouns or verbs.
When you "rein something in", you are metaphorically pulling on the straps to make the something slow down. YOU DO NOT "REIGN" SOMETHING IN. THIS MAKES NO SENSE.
Sorry for the rant, but I've seen this particular error half a dozen times today, and it's driving me batshit.