Summed Fraction Factorials

Algebra Level 5

If

\(\frac {3}{1! + 2! + 3!} + \frac {4}{2! + 3! + 4!} + \frac {5}{3! + 4! + 5!} + \ldots + \frac {2001}{1999!+ 2000! + 2001!} = \frac {a}{b} - \frac {c}{d!}\),

what is the digit sum of \((abcd)^{(abc)^{(ab)^a}}\)? Assume that \(a\)and \(b\); and \(c\) and \(d!\) are co-prime.

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