
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
$(1)$ Given $f(\sqrt{x}-1)=x+2\sqrt{x}$, find the analytical expression of $f(x)$ and indicate its domain;<br/>$(2)$ Find the range of the function $y=\frac{3-2{x}^{2}}{1+{x}^{2}}$;

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