An ideal gain contract multiplies the loaded node, not the open source voltage. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The gain stage receives the loaded node
The source is 6 volts, but the input node is only 3 volts after loading.
Vin=3V
The ideal gain multiplies the loaded node
The declared gain of 2 acts on the loaded input, giving 6 volts.