Maybe this is not Calculus

Calculus Level 3

\[ \large \lim_{n \to \infty } \frac{(n!)^{2}}{\displaystyle \sum_{k=1}^{n} \left(\frac{n!}{k} \right )^{2}} \]

If the value of the limit above is \( x\), and if the expression \(\displaystyle \frac { 4\left \lfloor \pi^{2}x \right \rfloor + 1 }{7 } \) can be expressed in the form \(\large \frac {a^{2}}{b} \) where \(a\) and \(b\) are coprime positive integers. Determine \(a + b\).

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