
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 $f(x) = \sqrt{3}\sin(\omega x + \varphi) + 2\sin^2\left(\frac{\omega x + \varphi}{2}\right) - 1$ ($\omega > 0$, $0 < \varphi < \pi$) is an odd function, and the distance between two adjacent axes of symmetry is $\frac{\pi}{2}$.
(1) When $x \in \left[-\frac{\pi}{2}, \frac{\pi}{4}\right]$, find the interval where $f(x)$ is strictly decreasing.
(2) The graph of the function $y = f(x)$ is translated right by $\frac{\pi}{6}$ units, then the horizontal coordinate is shrunk to half of its original value (the vertical coordinate remains unchanged), resulting in the graph of the function $y = g(x)$. When $x \in \left[-\frac{\pi}{12}, \frac{\pi}{6}\right]$, find the range of $g(x)$.

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