
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 $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $|\varphi| \leq \frac{\pi}{2}$), if $(-\frac{\pi}{4}, 0)$ is the center of symmetry of the graph of $f(x)$, and $x = \frac{\pi}{4}$ is an extremum point of $f(x)$, and $f(x)$ is monotonic in the interval $(\frac{5\pi}{18}, \frac{2\pi}{5})$, then the maximum value of $\omega$ is \_\_\_\_\_.

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