Please reason step by step, and put your final answer within \boxed{}.

Find all positive numbers \( x_{1}, x_{2}, \ldots, x_{10} \) that satisfy the condition \(\left( x_{1} + \ldots + x_{k} \right) \left( x_{k} + \ldots + x_{10} \right) = 1 \) for all \( k = 1, 2, \ldots, 10 \).
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