Lowest Value Function

Algebra Level 3

Define a function \(F\) as follows

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


\( S=aw+bx+cy+dz-\left| w-x \right| -\left| aw+bx-cy-\left| w-x \right| \right| \)
\(T=\left| aw+bx+cy-dz-\left| w-x \right| -\left| aw+bx-cy-\left| w-x \right| \right| \right| \)

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


Let \(a,b,c,d\) be integer digits of a \(4\) digit integer \(A = \overline{abcd} \). What is the value of \(A\)?


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