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The letter is legit. Hold on to it, pay the 180 and watch your credit. Or pay 180 through your bank account, once payment is accepted by Cox, download payment along with letter after copying and then send to credit bureaus. Do not call them. They may or may not call you back.
Just note that a collection account marked as "settled in full" will have no effect on your crefit score -- a settled collection (or a paid in full collection) hurts as much as the unpaid collection.
If it's outside the SOL, I would keep trying for PFD.
To err is human.
By calling and getting their review, they corrected an error.
A binding contract requires a clear meeting of the minds. They have corrected and provided a revised offer.
In my opinion, you cannot establish a binding contract when now aware of their error and revised offer.
You are on tenuous legal ground asserting a contractual agreement when advised that a prior offer was erroneous, and they have corrected.