
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 $f(x)=2\cos (x-\frac{π}{3})\cos\ \ x+1$.  
$(1)$ Find the interval where $f(x)$ is monotonically increasing.  
$(2)$ Given $f(\alpha +\frac{π}{12})=\frac{7}{6}$, $\alpha \in (0$,$\frac{π}{2}$), find the value of $\sin (\frac{7π}{6}-2\alpha )$.

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