
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

Given the functions:
$$
f(x) = 
\begin{cases} 
1, & \text{if} \ |x| \leq 1 \\ 
0, & \text{if} \ |x| > 1 
\end{cases}
$$

$$
g(x) = 
\begin{cases} 
2 - x^2, & \text{if} \ |x| \leq 1 \\ 
2, & \text{if} \ |x| > 1 
\end{cases}
$$

Find \( f[f(x)] \), \( f[g(x)] \), \( g[f(x)] \), and \( g[g(x)] \).

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