Ellen baked $2$ dozen cupcakes of which half contained chocolate, two-thirds contained raisins, one-fourth contained chocolate chips, and one-sixth contained nuts.  What is the largest possible number of cupcakes that had none of these ingredients?
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Ellen baked $2$ dozen cupcakes of which half contained chocolate, two-thirds contained raisins, one-fourth contained chocolate chips, and one-sixth contained nuts.  What is the largest possible number of cupcakes that had none of these ingredients?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8
Ellen baked $2$ dozen cupcakes of which half contained chocolate, two-thirds contained raisins, one-fourth contained chocolate chips, and one-sixth contained nuts.  What is the largest possible number of cupcakes that had none of these ingredients?
train