
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
$(1)$ Find the solution set of the inequality $x(x+2) \gt x(3-x)+6$;<br/>$(2)$ Find the domain of the function $y=\sqrt{x+1}+\frac{{{{(x-1)}^0}}}{{\sqrt{-{x^2}-x+6}}}$.

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