
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 $f(x)=2\sin x\cos x-\sqrt{3}\cos 2x (x\in R)$

(1) If $f(\alpha) = \frac{1}{2}$, and $\alpha \in (\frac{5\pi}{12}, \frac{2\pi}{3})$, find the value of $\cos 2\alpha$.

(2) Let the maximum value of the function $f(x)$ on $[\frac{\pi}{4}, \frac{\pi}{2}]$ be $b$, and the function $f(x)$ is monotonically increasing on $[a\pi, b\pi] (a < b)$, find the minimum value of the real number $a$.

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