
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 $0 < \alpha < \frac{\pi }{4}$, $\beta$ is the smallest positive period of $f(x)=\cos (2x+\frac{\pi }{8})$, $\overrightarrow{a}=(\tan (\alpha +\frac{1}{4}\beta )-1)$, $\overrightarrow{b}=(\cos \alpha ,2)$, and $\overrightarrow{a}\cdot \overrightarrow{b}=m$, find the value of $\frac{2{{\cos }^{2}}\alpha +\sin 2(\alpha +\beta )}{\cos \alpha -\sin \beta }=\_\_\_\_\_\_\_\_\_\_.$

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