
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 $f(x)=2\sin\left(\omega x+\frac{\pi}{4}\right)$ ($\omega>0$), its graph is shifted to the right by $\frac{\pi}{4\omega}$ units to obtain the graph of $y=g(x)$. If $y=g(x)$ is an increasing function on the interval $\left[-\frac{\pi}{6},\frac{\pi}{4}\right]$, find the maximum value of $\omega$.

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