Determine the complex number $z$ satisfying the equation $3z+4i\bar{z}=1-8i$.  Note that $\bar{z}$ denotes the conjugate of $z$.
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<|im_start|>user
Determine the complex number $z$ satisfying the equation $3z+4i\bar{z}=1-8i$.  Note that $\bar{z}$ denotes the conjugate of $z$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
-5+4i
Determine the complex number $z$ satisfying the equation $3z+4i\bar{z}=1-8i$.  Note that $\bar{z}$ denotes the conjugate of $z$.
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