Lowest Value Function

Algebra Level 3

Define a function FF as follows

F(w,x,y,z)=18(ST)F(w,x,y,z)=\dfrac { 1 }{ 8 } \left( S-T \right)

where

S=aw+bx+cy+dzwxaw+bxcywx S=aw+bx+cy+dz-\left| w-x \right| -\left| aw+bx-cy-\left| w-x \right| \right|
T=aw+bx+cydzwxaw+bxcywxT=\left| aw+bx+cy-dz-\left| w-x \right| -\left| aw+bx-cy-\left| w-x \right| \right| \right|

For certain positive integers a,b,c,da,b,c,d and for all w,x,y,zw,x,y,z, this function F(w,x,y,z)F\left(w,x,y,z\right) always returns the lowest of the values w,x,y,zw,x,y,z.     \;\; For example

F(4,3,2,1)=3F(4,-3,2,1)=-3.

Let a,b,c,da,b,c,d be integer digits of a 44 digit integer A=abcdA = \overline{abcd} . What is the value of AA?

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