For a real number $x,$ let
\[f(x) = \max \{|x + 15|, |x + 12|, |x + 8|, |x - 1|, |x - 3|, |x - 19|\}.\]Find the minimum value of $f(x).$
<|im_start|>system
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<|im_start|>user
For a real number $x,$ let
\[f(x) = \max \{|x + 15|, |x + 12|, |x + 8|, |x - 1|, |x - 3|, |x - 19|\}.\]Find the minimum value of $f(x).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
17
For a real number $x,$ let
\[f(x) = \max \{|x + 15|, |x + 12|, |x + 8|, |x - 1|, |x - 3|, |x - 19|\}.\]Find the minimum value of $f(x).$
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