
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 vectors $\overrightarrow{a}=(\cos^2 \omega x - \sin^2 \omega x, \sin \omega x)$ and $\overrightarrow{b}=(\sqrt{3}, 2\cos \omega x)$, let the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b} (x \in \mathbb{R})$ have its graph symmetric about the line $x= \frac{\pi}{2}$, where $\omega$ is a constant and $\omega \in (0,1)$.

(Ⅰ) Find the expression for the function $f(x)$;

(Ⅱ) If the x-coordinates of all points on the graph of $y=f(x)$ are changed to $\frac{1}{6}$ of their original values, and then the resulting graph is shifted to the right by $\frac{\pi}{3}$ units (the y-coordinates remain unchanged) to obtain the graph of $y=h(x)$, and if the equation $h(x)+k=0$ has exactly one real solution in the interval $\left[0, \frac{\pi}{2}\right]$, find the range of the real number $k$.

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