
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 $f(x)=2\cos^2(\omega x)-1+2\sqrt{3}\cos(\omega x)\sin(\omega x)$ $(0 < \omega < 1)$, the line $x= \frac{\pi}{3}$ is an axis of symmetry for the graph of $f(x)$.

(1) Find the value of $\omega$;

(2) The graph of the function $y=g(x)$ is obtained by stretching the horizontal coordinates of the points on the graph of $y=f(x)$ by a factor of $2$ and then shifting it to the left by $\frac{2\pi}{3}$ units. If $g(2\alpha+ \frac{\pi}{3})= \frac{6}{5}$, $\alpha \in (0, \frac{\pi}{2})$, find the value of $\sin(\alpha)$.

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