
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 $f(x)=\sqrt{2}(\sin \omega x+\cos \omega x)(\omega > 0)$, the graph of the function $f(x)$ is shifted to the right by $\frac{\pi }{4\omega }$ units to obtain the graph of the function $g(x)$. If $y=g(x)$ is monotonically increasing on $\left[ -\frac{\pi }{6},\frac{\pi }{3} \right]$, when $\omega$ takes the maximum value, determine the equation of one of the axes of symmetry of the graph of the function $f(x)$.

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