
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 $$f(x)=\sin(ωx+φ)(ω>0,|φ|<\frac{π}{2})$$, the graph is shifted to the right by $$\frac{π}{6}$$ units. The resulting graph is symmetric about the y-axis, and $$f(\frac{π}{ω})=-\frac{1}{2}$$. Determine the analytic expression of function f(x) when ω takes the smallest value.

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