From the binomial expansion of \((1 + 0.2)^{1000} = \sum_{k = 0}^{1000}A_{k},\)

when \(A_{k} = \binom{1000}{k}(0.2)^{k},\)

find the value of \(k\) where the maximum value of \(A_{k}\) is reached.

Credit goes to an AIME problem.

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