
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 $g(x)$ defined only for $x$ between $-5$ and $5$ inclusive. The function is given by different segments:
\[ g(x) = 
  \begin{cases} 
  -x^2 + 1 & \text{for } -5 \leq x \leq -1 \\
  2x + 3 & \text{for } -1 < x < 2 \\
  -x + 4 & \text{for } 2 \leq x \leq 5
  \end{cases} \]
Determine how many values of $x$ satisfy $g(g(x)) = 1$.

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