Just Factorials around

\[\large S_n = 1! + 2! + 3! + \ldots + n! \]

Suppose for integer \(n>3\), we define \(S_n\) as above. If \(a_n\) is the maximum positive integer that satisfy the equation \( \frac{S_n}{24} = a_n + \frac{ \lambda}{24} \) for constant \(\lambda\), find the value of \(\lambda \).

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