Solution system P-46 via Helen A. Mironchick's approach for P-45 (ege23.doc)

The technique suggested  in 
http://kpolyakov.spb.ru/download/mea-2016-8.pdf
is still not very popular. Although examples of its application
to the P-45 and P-46 clearly demonstrate that working with 
couples is not always the dominant approach in terms of transparency and speed of obtaining results. The same goes for the post 
https://mapping-metod.blogspot.com/2019/03/blog-post.html
Moreover, all Helen Mironchick's strategies are pure mathematics,
where it is not necessary to understand how a deep result has been obtained
in order to successfully apply it, and often in various fields 

Original system

We utilize core technique developed in
 http://kpolyakov.spb.ru/download/mea-2016-8.pdf

I just believe that it is a successful follow up Helen Mironchick's 
original idea proposed by her in P-45 solution.
********************************
Now convert to equivalent
********************************
x1⊕x2=1
x3⊕x4=1
x5⊕x6=1
x7⊕x8=1 
x9⊕x10=1  
x11+x12=1  
(x1⊕x3)*(x4⊕x8)*(x2⊕x12) =0

Due to :-

x3⊕x4=1 attracts x3+x4 =1
x5⊕x6=1 attracts x5+x6 =1
x7⊕x8=1 attracts x7+x8 =1
x9⊕x10=1 attracts x9+x10 =1

System 01

x1⊕x2=1
x3⊕x4=1
x5⊕x6=1   has 2^5*3 = 32*3 =96 solutions
x7⊕x8=1 
x9⊕x10=1  
x11+x12=1  

****************************************
Now define number  of solutions
System 02
****************************************

x1⊕x2=1
x3⊕x4=1
x5⊕x6=1
x7⊕x8=1 
x9⊕x10=1  
x11+x12=1  
(x1⊕x3)*(x4⊕x8)*(x2⊕x12) =1

******************************************
Change the sequence  of equations
******************************************

x7⊕x8=1 
x8⊕x4=1  <= from last row
x4⊕x3=1
x3⊕x1=1  <= from last row
x1⊕x2=1 
x2⊕x12=1 <= from last row
x12+x11=1
x5⊕x6=1
x9⊕x10=1

 Final answer would be 96 - 12 =84