Let \(a_1, a_2, \ldots, a_n\) be a sequence of negative real numbers such that the sum of the sequence is \(-100\) and the sequence follows an arithmetic progression with a common difference \(d = -2\). Find the value of \(n\) given that the first term \(a_1 = -4\). Let's think step by step and output the final answer within \boxed{}.
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