
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 functions $f(x)=2x^{2}-1$ and $g(x)=\left\{\begin{array}{l}{2x-1, x\geq 0}\\{2-x, x<0}\end{array}\right.$, find:

$(1)$ The range of $g(x)$;

$(2)$ The expression of $f(g(x))$;

$(3)$ Solve the inequality $f(x) > g(x)$.

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