
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 vectors
$$
\begin{array}{l}
\boldsymbol{m}=(\sqrt{3} \sin \omega x, \cos \omega x), \\
\boldsymbol{n}=(\cos \omega x,-\cos \omega x),
\end{array}
$$
where $\omega>0$. The function $f(x)=\boldsymbol{m} \cdot \boldsymbol{n}$ has a minimum positive period of $\frac{\pi}{2}$.

(1) Find the value of $\omega$;

(2) Given a triangle $\triangle ABC$ with sides $a, b, c$ satisfying $b^2 = ac$, and the angle opposite side $b$ is $x$, if the equation $f(x) = k$ has two distinct real solutions, determine the range of the real number $k$.

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