It seems rational but is truly irrational

Calculus Level 4

\[\displaystyle \sum_{n = 0}^{\infty} \dfrac {(4n)! (1103 + 26390n)}{(n!)^4 396^{4n}}\]

has a value such that the closest integer to it is \(x\). If the actual value of it is \(\dfrac {a}{b \sqrt{c} \cdot \pi}\) where the fraction is in the lowest terms and \(c\) is prime, find the value of \(x + a + b + c\)?

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