
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 $f(x)=\sin \omega x (\omega > 0)$, translate the graph of this function to the left by $\dfrac{\pi}{4\omega}$ units to obtain the graph of the function $g(x)$. If the graph of $g(x)$ is symmetric about the line $x=\omega$ and is monotonically increasing in the interval $(-\omega,\omega)$, determine the value of $\omega$.

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