Problematic Product Series!

Let \( P = \ \large{\prod_{k = 2}^{k \rightarrow \infty}(1 + \frac{(-1)^{k+1}(k-1)!}{a_k}}) \)

where the sequence \( a_t \) is given by the interesting recurrence relation \(a_2 = 2 \) and \(a_{t+1} = (t+1)(a_t + (-1)^{t+1}(t-1)!) \) for all \(t \ge 2\).

What's the value of \( \large e^P \) where \(e \) is the base of the natural logarithm?

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