
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 vectors $\overrightarrow{a}=(\cos 2x,\sin 2x)$ and $\overrightarrow{b}=(\sqrt{3},1)$, and the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}+m$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x\in[0, \frac{\pi}{2}]$, the minimum value of $f(x)$ is $5$, find the value of $m$.

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