
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 $f(x)=-x^{2}+mx+1$, where $m\in R$.  
$(1)$ When $m=0$, find the maximum value of $f(x)$.  
$(2)$ If the function $h(x)=f(x)+2x$ is an even function, find the value of $m$.  
$(3)$ Let $g(x)=2\sin(x+\frac{π}{6})$. If for any $x_{1}\in \left[1,2\right]$, there exists $x_{2}\in \left[0,\pi \right]$ such that $g(x_{2})=f(x_{1})$, find the range of values for $m$.

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