
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
Consider the function $f(x)=\sin(\omega x + \varphi)$ where $\omega > 0$ and $|\varphi| < \frac{\pi}{2}$. Within a certain period of the function, the interval where the function is monotonically decreasing is $\left[\frac{5\pi}{12}, \frac{11\pi}{12}\right]$. The graph of $y=f(x)$ is first translated left by $\frac{\pi}{4}$ units, and then all the points on the graph have their x-coordinates halved (the y-coordinates remain unchanged). Denote the resulting function as $g(x)$.  
(1) Find the analytical expression of $g(x)$.  
(2) Suppose the sides $a$, $b$, and $c$ of $\triangle ABC$ satisfy $b^2 = ac$, and the angle opposite to side $b$ is $x$. If the equation $g(x)=k$ has two distinct real roots, find the range of values for the real number $k$.

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