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Given a function $f(x)=a{x}^6+b{x}^3+cx+8$, find real numbers $a$, $b$ and $c$, such that $\underset{0}{\overset{1}{\int }}f(x)\mathrm{d}x=\frac{35}{4}$, $f^{\prime }(0)=2$ and $f^{\prime }(1)=180$.