
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 $f(x) = \sqrt{3}\sin(\omega x + \varphi) - \cos(\omega x + \varphi)$ $(0 < \varphi < \pi)$ 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 monotonically decreasing;
2. The graph of the function $y=f(x)$ is translated right along the x-axis by $\frac{\pi}{6}$ units, and then the horizontal coordinate is shortened to half of its original length (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 the function $g(x)$.

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