
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 $$f(x)=2\sin(\omega x+\phi)$$ ($$\omega>0$$, $$0<\phi<\frac{\pi}{2}$$), $$f\left(-\frac{\pi}{4}\right)=0$$, $$f\left(\frac{\pi}{4}-x\right)=f\left(\frac{\pi}{4}+x\right)$$, and $$f(x)$$ is monotonic in the interval $$\left(\frac{\pi}{18}, \frac{2\pi}{9}\right)$$, find the maximum value of $$\omega$$.

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