Egyptian Fraction?

Let \( x, \space y,\) and \(z \) be positive real integer, with \( 1000< x <y <z<2000 \) and satisfy the condition \( \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{7} + \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z} + \dfrac{1}{45} = 1 \).

Given further that \(x\) is divisible by 42, find \( x+y+z\).

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