This is obviously a factually complex situation that requires review of all the documentation involved. But I don't think any lawyer would advise anyone to sue a "major health insurance company" for $14,000, especially when that insurer has paid several times that amount for related coverage. You need to see your own litigator to get a more personalized response.
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The first place I would look would be the insurance policy terms for appealing an adverse decision. Pay special attention to time deadlines.
I think that suing to recover $14 where the company paid $72 will be an uphill battle, and trying to do so without an attorney - - even more so.
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Check to see what your insurance policy says about denial of benefits. Your attorney may file a demand before you proceed to litigation.