
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
Shift the graph of the function $f(x)= \frac{1}{2}\sin (2x+\varphi)$ to the left by $\frac{\pi}{6}$ units, and then stretch the x-coordinates of each point on the graph by a factor of $2$ (the y-coordinates remain unchanged) to obtain a graph that is symmetric about $x= \frac{\pi}{3}$. Determine the minimum value of $|\varphi|$.

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