GAIC Math 2024 Problem 91

In the convex quadrilateral $A B C D$ inscribed in a circle, if $\overrightarrow{A B}+3 \overrightarrow{B C}+2 \overrightarrow{C D}+4 \overrightarrow{D A}=0$, and $|\overrightarrow{A C}|=4$, then the maximum of $|\overrightarrow{A B}|+|\overrightarrow{B C}|$ is _____.