
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 $f(x) = \sqrt{3}\sin\omega x \cdot \cos\omega x + \cos^2\omega x - \frac{1}{2}$, ($\omega > 0$), its smallest positive period is $\frac{\pi}{2}$.

(1) Find the expression for $f(x)$;  
(2) The graph of the function $f(x)$ is translated to the right by $\frac{\pi}{24}$ units, and then the x-coordinates of each point on the graph are stretched to four times their original length (the y-coordinates remain unchanged), resulting in the graph of the function $y=g(x)$. If the equation $g(x) + m = 0$ has exactly one real solution in the interval $[0, \frac{5\pi}{6}]$, find the range of the real number $m$.

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