
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 $f(x) = \sin \omega x - \cos \omega x$ with $\omega > \frac{2}{3}$, if the $x$-coordinate of the intersection of any axis of symmetry of the graph of $f(x)$ with the $x$-axis does not belong to the interval $(2\pi, 3\pi)$, determine the range of $\omega$. (Present the result as an interval).

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