
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 $f(x)=\sin(\omega x+\varphi)$ ($\omega > 0$, $|\varphi| < \frac{\pi}{2}$) whose graph passes through the point $(0,\frac{1}{2})$, if $f(x) \leqslant f\left( \frac{\pi}{12} \right)$ holds true for all $x \in \mathbb{R}$, determine the minimum value of $\omega$.

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