2003 IMO Shortlist S09

Let $x_0, x_1, x_2, \ldots$ be the sequence defined by $x_i= 2^i$ if $0 \leq i \leq 2003$ $x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$ Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.

设 $x_0, x_1, x_2, \ldots$ 为由 $x_i= 2^i$ 定义的序列 if $0 \leq i \leq 2003$ $x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$ 找到序列中包含可被 2004 整除的 $k$ 个连续项的最大 $k$。