
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 $f\left(x\right)=2\sin \left(2x+\varphi \right)\left(0 \lt \varphi\ \ \lt \pi \right)$, and $\forall x\in R$, $f(\frac{π}{6}+x)=f\left(-x\right)$.  
$(1)$ Find the set of values of $x$ when $f\left(x\right)$ reaches its maximum value.  
$(2)$ If there exists a unique real number $x_{0}\in \left[0,m\right]$, such that $f(x_{0})=-\sqrt{2}$, find the range of real number $m$.

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