
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 $f(x)=\sin \left( 2x-\frac{\pi }{6} \right)+2\cos^2x-1$.

(Ⅰ) Find the maximum value of the function $f(x)$ and the corresponding set of values for $x$;

(Ⅱ) If $\frac{\pi }{4} < \alpha < \frac{\pi }{2}$ and $f(\alpha)=\frac{4}{5}$, find the value of $\cos 2\alpha$.

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