
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 $f\left( x \right)=\sin \left( \omega x+\varphi \right)\left( \omega > 0,\left| \varphi \right|\leqslant \dfrac{\pi }{2} \right)$, and knowing that $x=-\dfrac{\pi }{4}$ is a zero point of $f\left( x \right)$, $x=\dfrac{\pi }{4}$ is the axis of symmetry of the graph of $y=f\left( x \right)$, and $f\left( x \right)$ is monotonic in $\left( \dfrac{\pi }{18},\dfrac{5\pi }{36} \right)$, find the maximum value of $\omega$.

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