
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 $f(x) = \sin^2 \omega x - \frac{1}{2}$ ($\omega > 0$) has a period of $\frac{\pi}{2}$. Find the minimum value of the real number $a$ if its graph is translated to the right by $a$ units ($a > 0$) along the x-axis, and the resulting graph is symmetric about the origin.

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