
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 graph of a function $f(x)$ is given by parts: for $-4 \leq x \leq -2$, $f(x) = -x^2 - 2x + 6$; for $-2 < x \leq 2$, $f(x) = x + 4$; and for $2 < x \leq 4$, $f(x) = x^2 - 4x + 8$. Determine how many values of $x$ satisfy $f(f(x)) = 4$ within the range $x \in [-4,4]$.

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