
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 $f(x) = 4\cos\left(\frac{\pi}{3} - \omega x\right)\cos(\omega x) - 1$ ($\omega > 0$), the distance between two adjacent axes of symmetry of its graph is $\frac{\pi}{2}$.
1. Find the analytical expression of the function $f(x)$.
2. Find the intervals of monotonic increase of $f(x)$ on $[0, 2\pi]$.

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