
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 vectors $a=(\sin(\omega x), \cos(\omega x))$ and $b=(\cos(\phi), \sin(\phi))$, the minimum positive period of the function $f(x)=a \cdot b$ is $2\pi$ under the conditions $\omega > 0$ and $\frac{\pi}{3} < \phi < \pi$. The graph of the function passes through point $M(\frac{\pi}{6}, \frac{\sqrt{3}}{2})$.

(1) Find the analytic expression of function $f(x)$;
(2) Given $\alpha, \beta \in (0, \frac{\pi}{2})$ and $f(\alpha)= \frac{3}{5}$, $f(\beta)= \frac{12}{13}$, find the value of $f(2\alpha-\beta)$.

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