
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
If the set of function values of $y=f\left(x\right)$ on the interval $\left[m,n\right]$ is exactly $[{\frac{1}{n},\frac{1}{m}}]$, then the interval $\left[m,n\right]$ is called a "$\Gamma $ interval" of $y=f\left(x\right)$. Let $f\left(x\right)=-x|x|+2x$.<br/>$(1)$ If the function $y=f\left(x\right)$ is strictly increasing on interval $A$, please directly write down the interval $A$ (one is enough);<br/>$(2)$ Determine whether the interval $[{\frac{1}{2},\frac{3}{2}}]$ is a "$\Gamma $ interval" of the function $y=f\left(x\right)$, and explain the reason;<br/>$(3)$ Find the "$\Gamma $ interval" of the function $y=f\left(x\right)$ within $\left[1,+\infty \right)$.

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