
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

The function \( J(x) \) is defined by:
\[ 
J(x)=\left\{\begin{array}{ll}
4+x & \text{ for } x \leq -2, \\
-x & \text{ for }-2 < x \leq 0, \\
x & \text{ for } x > 0.
\end{array}\right.
\]

How many distinct real solutions does the equation \( J(J(J(x))) = 0 \) have?

Present the answer in LaTex format: \boxed{Your answer}
user