# Can you simplify this otherwise tough problem??

Algebra Level 4

The number of distinct $$4$$-tuples of rational numbers $$(a,b,c,d)$$ satisfying $a \log 2+ b \log 3 + c \log 5 + d \log 7=2012$ is equal to $$k$$. Enter your answer as the sum of $$k$$ and all possible values of the $$4$$-tuples.

Details: Logarithm is taken to base $$10$$.

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

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