
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 a real number \(\omega > 0\), suppose the function \(f(x) = \sin^2 (\omega x) + \sqrt{3} \sin (\omega x) \cdot \sin \left(\omega x + \frac{\pi}{2}\right)\) has the smallest positive period of \(\frac{\pi}{2}\). Find the maximum and minimum values of \(f(x)\) on the interval \(\left[\frac{\pi}{8}, \frac{\pi}{4}\right]\).

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