GAIC Math 2024 Problem 85

In the Cartesian coordinate plane $x O y$, point $A(a, 0), B(0, b), C(0,4)$, moving point $D$ satisfies $|C D|=1$. If the maximum value of $|\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O D}|$ is 6, then the minimum value of $a^{2}+b^{2}$ is _____.