Given

\[\begin{cases}a,b,c,d>0\\ab+bc+cd+da=1,\end{cases}\]

\(k\) is the largest number such that

\[\displaystyle \frac{a^3+c^3}{2\sqrt{bd}}+\frac{b^3+d^3}{2\sqrt{ac}}\ge k.\]

Equality is reached when \(a=a_{\text{eq}}\), \(b=b_{\text{eq}}\), \(c=c_{\text{eq}}\), \(d=d_{\text{eq}}\).

Find \(a_{\text{eq}}+b_{\text{eq}}+c_{\text{eq}}+d_{\text{eq}}+k\) and round it to \(3\) decimal places.

Original.

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