
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 $f(x) = \sqrt{1-x}$, $g(x) = \sin x \cdot f(\sin 2x) + \frac{\sqrt{6} + \sqrt{2}}{4}f(\cos 4x)$, where $x \in \left[-\frac{\pi}{4}, 0\right]$

(1) Simplify the function $g(x)$ into the form $A\sin(\omega x + \varphi) + B$ ($A, B \in \mathbb{R}, \omega > 0, \varphi \in (-\pi, \pi)$).

(2) Find the range of the function $g(x)$.

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