
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
Let $f(x) = 4\cos(wx+\frac{\pi}{6})\sin(wx) - \cos(2wx) + 1$, where $0 < w < 2$.
1. If $x = \frac{\pi}{4}$ is a symmetry axis of the function $f(x)$, find the period $T$ of the function.
2. If the function $f(x)$ is increasing on the interval $[-\frac{\pi}{6}, \frac{\pi}{3}]$, find the maximum value of $w$.

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