
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 $f(x) = 2 - 2\cos^2\left(\frac{\pi}{4} + x\right) - \sqrt{3}\cos(2x)$,  
(1) Find the intervals of increase for $f(x)$ when $x \in [0, \pi]$.  
(2) Find the axis of symmetry for $f(x)$.  
(3) If the equation $f(x) - k = 0$ has solutions in $x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the range of the real number $k$.

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