
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 function $f\left(x\right)=2e^{2x}-a\left(2x+1\right)\ln \left(2x+1\right)$.<br/>$(1)$ When $a=2$, investigate the monotonicity of the function $f\left(x\right)$;<br/>$(2)$ When $x∈[{0,\frac{π}{2}}]$, $f\left(x\right)\geqslant 2\cos ^{2}x-2\left(a-2\right)x$ always holds, find the range of values for $a$.

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