
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 universal set $U = \mathbb{R}$, $A = \{x | -4 \leq x < 2\}$, $B = \{x | 0 < x + 1 \leq 4\}$, $P = \{x | x \leq 0$ or $x \geq 5\}$.  
$(1)$ Find $A \cap B$, $\complement_{U}B$;  
$(2)$ Find $(A \cap B) \cup (\complement_{U}P)$.

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