
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
The graph of the function $f(x)=2\sin(2\omega x-\frac{π}{3})(\omega>0)$ is shifted to the left by $\frac{π}{6\omega}$ units to obtain the graph of the function $y=g\left(x\right)$. If $y=g\left(x\right)$ is increasing on $[0,\frac{π}{4}]$, then the maximum value of $\omega$ is ______.

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