
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 $\overrightarrow{a}=(\sin 2x, 2m)$, $\overrightarrow{b}=(\sqrt{3}, \cos^2 x)$, and the function $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}-2m$.  
$(1)$ When $m=0$, find the monotonic increasing interval of $f(x)$.  
$(2)$ If the maximum value of the function $f(x)$ on $\mathbb{R}$ is $1$, find the set of values of $x$ for which $f(x) \geq 0$.

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