The complex number $$\(1+2 \mathrm{i}\)$$ is denoted by $$\(u\)$$. The polynomial $$\(2 x^{3}+a x^{2}+4 x+b\)$$, where $$\(a\)$$ and $$\(b\)$$ are real constants, is denoted by $$\(\mathrm{p}(x)\)$$. It is given that $$\(u\)$$ is a root of the equation $$\(\mathrm{p}(x)=0\)$$. Find the least value of $$\(\operatorname{Im} z\)$$ for points in the shaded region. Give your answer in an exact form. ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................ ................................................................................................................................................

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