
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 $A > 0,\omega > 0,\text{ and }|\varphi| < \frac{\pi }{2}$, and it satisfies $f(x+\frac{\pi }{2})=-f(x-\frac{\pi }{2})$ and $f(\frac{\pi }{6}+x)=f(\frac{\pi }{6}-x)$, determine the expression for $f(x)$.

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