When power is fed into the Fujitsu unit, neither the AC unit's thermostat nor gateway are powered

Check the wiring and that the polarity is correct:
- Gateway-AC unit thermostat connection. Polarity: Red-Red, Grey-White and Black-Black.
- Gateway-indoor unit connection. Polarity: Red-Y1, Grey-Y2 and Black-Y3.

If the wiring is done poorly, the electronics may be damaged, particularly if the red wire is connected to the black wire, since we would be feeding power inversely to both the gateway and thermostat, positive with negative and negative with positive, which could cause damage.
If either the black or red wire is interchanged with the grey/white cable, the effect would be of no power to any of the devices and in principle this should not affect the electronics, but it is better not to do this.