
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 $f(x)=\frac{1}{2}\sin (\omega x+\frac{\pi }{6})$ ($\omega > 0$), its graph is tangent to the line $y=m$. The distance between adjacent tangent points is $\pi$. If point $A({x}_{0},{y}_{0})$ is a symmetry center of the graph $y=f(x)$ and ${x}_{0}\in [0,\frac{\pi }{2}]$, find the value of ${x}_{0}$.

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