
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 tangent function $f(x) = \tan(\omega x + \frac{\pi}{4})$ with $\omega > 0$, and $f(x)$ is monotonically increasing on the interval $(-\frac{\pi}{3}, \frac{\pi}{2})$, find the maximum value of $\omega$.

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