
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 sets $A=\{x \mid x+2\geq 0, x\in \mathbb{R}\}$ and $B=\left\{x \bigg| \frac{x-1}{x+1} \geq 2\right\}$.

(1) Find the intersection $A \cap B$ and the union $A \cup B$ of sets $A$ and $B$.

(2) Find the intersection of the complement of $A$ (in the universal set $U$) with $B$, denoted $(\complement_U A) \cap B$.

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