Confusion about apply nodal analysis
The verbal description of the circuit sounds like this:







simulate this circuit &ndash; Schematic created using CircuitLab

Unfortunately, this circuit is unsolvable. VCVS1, configured with gain 1, is asked reproduce the voltage present across R1, while simultaneously being forced to 12V plus the voltage across R1. VCVS1 needs non-unity gain in order for this to be solvable.