
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 the function $f(x)$, as shown, is modified so that now we need to determine how many values of $x$ satisfy $f(f(x)) = 5$. Consider the modified segments as follows:
1. $f(x) = -x + 6$ when $-4 \leq x \leq -1$.
2. $f(x) = x - 1$ when $-1 < x \leq 3$.
3. $f(x) = -2x + 10$ when $3 < x \leq 5$.

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