Solving Many Equations At Once

Algebra Level 3

\[x_1 + x_2 + x_3 + \ldots + x_{1007} = 2014^2 \\ \frac {x_1}{x_1 + 1} = \frac {x_2}{x_2 + 3} = \frac {x_3}{x_3 + 5} = \ldots = \frac {x_{1007}}{x_{1007} + 2013} \]

Real numbers \(x_{1}, x_{2}, x_{3}, ..., x_{1007}\) satisfy the conditions above. Find the value of \(x_{253}\).

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