Senior Kangaroo 2019 Q8

The function $J\left(x\right)$ is defined by:

$J\left(x\right)=\begin{cases} 4+x & \text{for }x\le-2 \\ -x & \text{for }-2 < x\le0 \\ x & \text{for } x>0.\end{cases}$

How many distinct real solutions has the equation $J\left(J\left(J\left(x\right)\right)\right)=0$?