Challenging Sum

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009:

“Four positive integers a, b, c, and d are such that
_________abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d = 2009.
What is the value of a + b + c + d?
_________A 73_________B 75_________C 77_________D 79_________E 81”

See the Challenging Sum

(Update 4/17/2019) I forgot to mention in my write-up that the UKMT solution only included the last few steps and did not provide any motivation for why one would think of it.

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