
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 that the power function $f(x)=({{m^2}-2m+2}){x^{5k-2{k^2}}}({k∈Z})$ is an even function and monotonically increasing on $\left(0,+\infty \right)$.
$(1)$ Find the analytical expression of the function $f\left(x\right)$;
$(2)$ If $f\left(2x-1\right) \lt f\left(2-x\right)$, find the range of values for $x$;
$(3)$ If positive real numbers $a$ and $b$ satisfy $a+b=4m$, find the minimum value of $\frac{1}{{a+1}}+\frac{4}{{b+1}}$.

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