This 2013 F150 PDF for CONVENTIONAL AND 5th-WHEEL TOWING towing guide
The pdf has the F150 Supercrew 4x4 5.0L 4-Valve V8 with 3.55 axle ratio with a maximum max. trailer weight of 7,700 lbs
5.0L - GCWR -12,900, 13,300, 13,500, & 14,900 (max.) lbs.
The GVWR isn't your actual towing capacity, it's used to estimate a vehicles towing capacity along with other ratings.
I was told the GVWR set by the manufacture can only be changed by the manufacture.
To actually change a vehicles GVWR from a door sticker, you would need to obtain a new weight certificate from the manufacture.
I came across a SUV where the GVWR (set by the manufacture) is significantly less than the auto manufactures a standard and optional tow capacity ratings. The SUV has a max. 2000 lb. standard TCR and an optional TCR of 5000 lbs max., with an automatic transmission. Using the vehicle door jam sticker GVWR the SUV's tow capacity would be much less than the towing capacity set by the manufacture, not matter the tow vehicles payload and trailers TW.
Essentially the GVWR is based on a fully loaded vehicle, minus any towing. A fully loaded vehicle may include both a payload and TW.
There are tow capacity calculators that use a vehicles GVWR along with other input to "estimate" towing capacity.
I'm have always been well under the manufactures tow capacity rating and haven't needed to think and worry much about a vehicles payload and towing capacity.
Towing add-ons and re-gearing axles may help facilitate towing but won't actually change any ratings set by a manufacture, unless the manufacture provides new ratings with door sticker for a particular vehicle.