
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 \( f(x) \) be a piece-wise linear function defined by three pieces as follows: \( f(x) = x + 2 \) when \( x \leq -1 \), \( f(x) = 5 - x \) when \( -1 < x \leq 3 \), \( f(x) = x - 4 \) when \( x > 3 \). How many values of \( x \) satisfy \( f(f(x)) = 0 \)?

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