A Large Harmonic Mean

Algebra Level 4

The harmonic mean of the following numbers:

\[1 \times 2 \times 3 \times 4 \times \ldots \times 2013, \\ 1 \times 2 \times 3 \times \ldots \times 2012 \times 2014, \\ 1 \times 2 \times \ldots \times 2011 \times 2013 \times 2014, \\ \vdots \\ 2 \times 3 \times 4 \times \ldots \times 2014 \]

can be expressed in the form \(\frac{a}{b} 2014!,\) where \(a\) and \(b\) are coprime integers. Find \(a+b\).

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