
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
$$
f(x) = 2 \sin^2\left(\frac{\pi}{4} + x\right) - \sqrt{3} \cos 2x - 1,
$$

where $x$ is in the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$.

(1) Find the interval where $f(x)$ is monotonically increasing.
(2) If the inequality $|f(x) - m| < 2$ holds for $x$ in the interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, determine the range of the real number $m$.

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