If the area bounded by circle \( x^{2}+y^{2}=4 \), then parabola \( y=x^{2}+x+1 \) and the curve....

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If the area bounded by circle \( x^{2}+y^{2}=4 \), then
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parabola \( y=x^{2}+x+1 \) and the curve \( y=\left[\sin ^{2} \frac{x}{4}+\cos \frac{x}{4}\right] \), (where [ ] denotes the greatest integer function) and \( x \)-axis \( \left(\sqrt{3}+\frac{2 \pi}{3}-\frac{1}{k}\right) \), then the numerical quantity \( k \) should be:


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