Let \( N > 1000 \) be an integer. Define a sequence \( A_n \) by

\[

A_0 = 1, \, A_1 = 0, \, A_{2k+1} = \frac{2k}{2k+1} A_{2k} + \frac{1}{2k+1} A_{2k-1}, \, A_{2k} = \frac{2k-1}{2k} \frac{A_{2k-1}}{N} + \frac{1}{2k} A_{2k-2}.

\]

Show that the following inequality holds for any integer \( k \) with \( 1 \leq k \leq (1/2) N^{1/3} \).

\[

A_{2k-2} \leq \frac{1}{\sqrt{(2k-2)!}}.

\]

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