
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 $\overrightarrow{m}=(-1,\cos \omega x+ \sqrt {3}\sin \omega x)$ and $\overrightarrow{n}=(f(x),\cos \omega x)$, where $\omega > 0$, and $\overrightarrow{m}\perp \overrightarrow{n}$. Also, the graph of the function $f(x)$ has a distance of $\frac {3}{2}\pi$ between any two adjacent axes of symmetry.  
(Ⅰ) Find the value of $\omega$;  
(Ⅱ) Let $\alpha$ be an angle in the first quadrant, and $f\left( \frac {3}{2}\alpha+ \frac {\pi}{2}\right)= \frac {23}{26}$, find the value of $\frac {\sin (\alpha+ \frac {\pi}{4})}{\cos (4\pi +2\alpha )}$.

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