Just Factorials around

Sn=1!+2!+3!++n!\large S_n = 1! + 2! + 3! + \ldots + n!

Suppose for integer n>3n>3, we define SnS_n as above. If ana_n is the maximum positive integer that satisfy the equation Sn24=an+λ24 \frac{S_n}{24} = a_n + \frac{ \lambda}{24} for constant λ\lambda, find the value of λ\lambda .

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