
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
The function $f(x) = \sqrt{3}\cos^2(\omega x + \varphi) - \cos(\omega x + \varphi) \cdot \sin(\omega x + \varphi + \frac{\pi}{3}) - \frac{\sqrt{3}}{4}$ ($\omega > 0$, $0 < \varphi < \frac{\pi}{2}$) satisfies the following two conditions:  
① The extremal points of the graph of $f(x)$ and the two symmetry centers adjacent to it form an isosceles right triangle.  
② $(\frac{2}{3}, 0)$ is a symmetry center of $f(x)$.  
(1) When $x \in [0, 2]$, find the intervals where $f(x)$ is monotonically decreasing.  
(2) Let $g(x) = f^2(x - \frac{5}{6}) + \frac{1}{4}f(x - \frac{1}{3}) + m$. If $g(x)$ has a root when $x \in [\frac{5}{6}, \frac{3}{2}]$, find the range of values for $m$.

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