
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
What is the smallest value of  $k$  for which the inequality
\begin{align*}
ad-bc+yz&-xt+(a+c)(y+t)-(b+d)(x+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  $a,b,c,d,x,y,z,t$ ?

Edit: Fixed a mistake! Thanks @below.

Present the answer in LaTex format: \boxed{Your answer}
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