
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 $f(x) = \sin(\omega x + \varphi)$ (where $\omega > 0$, $|\varphi| < \frac{\pi}{2}$), if the graph of the function is shifted to the left by $\frac{\pi}{12}$ units and the resulting graph is symmetric about the y-axis, and if the graph of the function is shifted to the right by $\frac{\pi}{6}$ units and the resulting graph is symmetric about the origin, determine the value of $\omega$ that satisfies the given conditions.

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