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The side lengths of an equiangular octagon are $1$, $2$, $3$, $4$, $1$, $2$, $3$, and $4$ in clockwise order. Find the octagon's area. [asy] size(150); pair A,B,C,D,EE,F,G,H,WW,X,Y,Z; A=(0,0); B=A+1*dir(0); C=B+4*dir(45); D=C+3*dir(90); EE=D+2*dir(135); F=EE+1*dir(180); G=F+4*dir(225); H=G+3*dir(270); WW=extension(G,H,A,B); X=extension(D,C,A,B); Y=extension(C,D,EE,F); Z=extension(EE,F,G,H); draw(A--B--C--D--EE--F--G--H--A); label( "$1$" , A--B , S ) ; label( "$1$" , EE--F , N ) ; label( "$3$" , C--D , E ) ; label( "$3$" , G--H , W ) ; label( "$4$" , C--B , NW ) ; label( "$4$" , G--F , SE ) ; label( "$2$" , D--EE , SW ) ; label( "$2$" , H--A , NE ) ; [/asy] Enter your answer in the form $x+y\sqrt z$ in simplest radical form.

 Feb 2, 2022
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The area is 10 + 14*sqrt(2).

 Feb 2, 2022

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