
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 $\overrightarrow{a} = (\sqrt{2}\sin(\frac{\pi}{4}+mx), -\sqrt{3})$ and $\overrightarrow{b} = (\sqrt{2}\sin(\frac{\pi}{4}+mx), \cos 2mx)$ where $x \in \mathbb{R}$, $m \in \mathbb{R}$, and the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$,

(Ⅰ) When $m=1$, and $x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the maximum and minimum values of $f(x)$;

(Ⅱ) When $m= \frac{n\pi}{2}$, if $f(x)$ has exactly $2017$ zeros in the interval $[0,2017]$, find all possible integer values of $n$.

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