
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
The function $f(x)=\cos(\omega x + \frac{\pi}{3})$ (where $\omega > 0$) has a property that the distance between any two adjacent axes of symmetry on its graph is $\frac{\pi}{2}$.  
(1) Determine the intervals of monotonicity and the center of symmetry for the function $f(x)$.  
(2) If the equation $2\cos^2x + m\cos x + 2=0$ has real solutions for $x \in (0, \frac{\pi}{2})$, find the range of the real number $m$.

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