Let a be a real number. If the value of definite integral \( \int_{...

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Let a be a real number. If the value of definite integral \( \int_{-\pi+a}^{3 \pi+a}|x-a-\pi| \sin \left(\frac{x}{2}\right) d x \) is
\( \mathrm{P} \)
W equal to -16 then sum of all the values of a in \( [0,314] \) is \( k \pi \). Find the value of \( k \).
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