
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 $f(1+x)+2f(1-x)=6-\frac{1}{x}$, find the value of $f(\sqrt{2})$;<br/>$(2)$ The power function $g(x)=({m}^{2}-2m-2){x}^{{m}^{2}+3m+2}$ is monotonically increasing on $\left(0,+\infty \right)$, if $g\left(2x-1\right)\geqslant 1$, find the range of values for $x$.

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