Joachim's regular tax liability calculated on his $2023$ Form $1040$ is $$451,890.$ Which of the following statements regarding the Alternative Minium Tax $($AMT$)$ is TRUE?

If Joachim's tentative minimum tax is $$421,200,$ his total tax liability is $$421,200.$

If Joachim's AMT is $$10,112,$ his total tax liability is $$451,890.$

If Joachim's AMT is $$6,380,$ his total tax liability is $$458,270.$

If Joachim's tentative minimum tax is $$500,700,$ his total tax liability is $$952,590.$