
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 $f(x) = \sin \left(2x- \frac{\pi}{3}\right) + \cos \left(2x- \frac{\pi}{6}\right) + 2\cos^2 x-1$.

(1) Find the smallest positive period of the function $f(x)$.

(2) If $\alpha \in \left[ \frac{\pi}{4}, \frac{\pi}{2}\right]$ and $f(\alpha) = \frac{3 \sqrt{2}}{5}$, find $\cos 2\alpha$.

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