
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 $f(x)=\cos(2x+\frac{π}{3})+\sin^{2}x-\cos^{2}x+2\sqrt{3}\sin x\cos x$.  
$(1)$ Express the function in terms of sine function.  
$(2)$ If $f(\alpha)=\frac{1}{7}$ and $2\alpha$ is an angle in the first quadrant, find $\sin 2\alpha$.

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