\[ \large \begin{cases} p!+1= (2p+1)^{2} \\ q!+1=(10q+p-4)^{2} \end{cases} \]

Let \(p\) and \(q\) be prime numbers satisfying the system of equations above. Find \(p + q\).

**Hint**: You may find Wilson's theorem useful.

**Notation**: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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