Find the minimum value of
\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\]over all real numbers $x,$ $y,$ and $z.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the minimum value of
\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\]over all real numbers $x,$ $y,$ and $z.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-14
Find the minimum value of
\[3x^2 + 12y^2 + 27z^2 - 4xy - 6xz - 12yz - 8y - 24z\]over all real numbers $x,$ $y,$ and $z.$
train