
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 the power function $f(x)=({3{m^2}-2m+1}){x^{3k-{k^2}+4}}(k∈{Z})$ is an even function and monotonically increasing on $\left(0,+\infty \right)$.
$(1)$ Find the analytical expression of the function $f\left(x\right)$;
$(2)$ Solve the inequality $f\left(3x+2\right) \gt f\left(1-2x\right)$.

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