What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(x+t)-(b+d)(y+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*}holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?

What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(x+t)-(b+d)(y+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*}holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?

What is the smallest value of $k$ for which the inequality \begin{align*} ad-bc+yz&-xt+(a+c)(x+t)-(b+d)(y+z)\leq \\ &\leq k\left(\sqrt{a^2+b^2}+\sqrt{c^2+d^2}+\sqrt{x^2+y^2}+\sqrt{z^2+t^2}\right)^2 \end{align*}holds for any $8$ real numbers $a,b,c,d,x,y,z,t$?

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