Can you simplify this otherwise tough problem??

Algebra Level 4

The number of distinct 44-tuples of rational numbers (a,b,c,d)(a,b,c,d) satisfying alog2+blog3+clog5+dlog7=2012a \log 2+ b \log 3 + c \log 5 + d \log 7=2012 is equal to kk. Enter your answer as the sum of kk and all possible values of the 44-tuples.

Details: Logarithm is taken to base 1010.

Example: If k=2k=2 and the solutions are (1,2,3,4),(5,6,7,8)(1,2,3,4), (5,6,7,8) then your answer should be (1+2+3+4)+(5+6+7+8)+2.(1+2+3+4)+(5+6+7+8)+2.

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