
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 $f(x) = 2\sin(\omega x + \varphi - \frac{\pi}{6})$ with $0 < \varphi < \pi$ and $\omega > 0$ is an even function, and the distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac{\pi}{2}$,  
(1) Find the range of $f(x)$ when $x \in \left[\frac{\pi}{6}, \frac{5\pi}{6}\right]$;  
(2) Translate the graph of the function $y = f(x)$ to the right by $\frac{\pi}{6}$ units, and then stretch the x-coordinates of the resulting graph by a factor of 4 while keeping the y-coordinates unchanged, obtaining the graph of the function $y = g(x)$. Find the intervals where $y = g(x)$ is decreasing on $x \in [0, 4\pi]$.

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