
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 $f(x) = \sin(2x + \frac{\pi}{3}) + \cos(2x - \frac{\pi}{6}), (x \in \mathbb{R})$.
(I) Find the intervals on which $f(x)$ is strictly increasing.
(II) If $f\left(\frac{\alpha}{2} - \frac{\pi}{6}\right) = \frac{6}{5}$, where $\alpha \in \left(\frac{\pi}{2}, \pi\right)$, find the value of $\tan\left(\alpha - \frac{\pi}{4}\right)$.

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