
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 $\overrightarrow{a} = (\sin x, -\cos x)$, $\overrightarrow{b} = (\sqrt{3}\cos x, \cos x)$, let the function $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - \frac{1}{2}$;
(1) Determine the intervals on which the function $f(x)$ is increasing;
(2) If $x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the maximum and minimum values of the function $f(x)$ and the corresponding values of $x$;
(3) If the inequality $|f(x) - m| < 1$ holds for all $x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, find the range of the real number $m$.

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