
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 $f(x) = \sin (\omega x + \phi)$ $(\omega > 0, -\frac{\pi}{2} \leqslant \phi \leqslant 0)$, $x = -\frac{\pi}{4}$ is a zero of $f(x)$, $x = \frac{\pi}{4}$ is the axis of symmetry for the graph of $y = f(x)$, and $f(x)$ is monotonic in the interval $\left( \frac{\pi}{18}, \frac{5\pi}{36} \right)$, determine the number of possible values for $\omega$.

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