What is the average value of the square of the maximum of a normalized vector?

Question

Consider a normalized vector $\textbf{v}$ of length $N$, that therefore belongs to the surface of an $N$-sphere. Take now the maximum of the absolute value of the vector. For example for $N$=2, this function is $\max(|\cos(\theta)|,|\sin(\theta)|)$. Consider now the square of this value. Is there a smart way to calculate the average of this value over the $N$-sphere? I can do it for $N$=2 and it equals $\frac{\frac{\pi}{2}+1}{\pi}$ but in general it seems challenging. Thanks in advance.