
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 $f\left(x\right)=4\cos x\cdot \sin (x+\frac{π}{6})-1$.  
$(1)$ Find the period of $f\left(x\right)$;  
$(2)$ If $f\left(\alpha \right)=\frac{6}{5}$, where $\alpha \in (0$,$\frac{π}{4})$, find $\cos 2\alpha$.

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