
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
Consider the function $f(x)$ defined on the interval $[-5, 5]$ by the following rules: for $-5 \leq x \leq -1$, $f(x) = -\frac{1}{2}x^2 - 2x + 1$; for $-1 < x \leq 2$, $f(x) = x + 1$; for $2 < x \leq 5$, $f(x) = \frac{1}{3}x^2 - 2x + 4$. How many values of $x$ satisfy $f(f(x)) = 4$?

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