
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 $U = \mathbb{R}$, $M = \{ x | y = \sqrt{x - 2} \}$, $N = \{ x | x < 1 \text{ or } x > 3 \}$. Find: $(1)$ The set $M \cup N$; $(2)$ The set $M \cap (\complement_U N)$.

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