
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
Let \( g(x) \) be defined as:
\[
g(x) = \begin{cases}
x^2-3 & \text{if } x \ge -3, \\
x + 4 & \text{if } -6 \le x < -3, \\
-2x & \text{otherwise},
\end{cases}
\]
Determine for how many distinct values of \( x \) the equation \( g(g(x)) = 4 \) holds.

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