As \(x\) and \(y\) range over all non-zero real values, what is the minimum value of \[ \frac { |x+y| } { | x | + | y | }? \]

**Details and assumptions**

The notation \( | \cdot | \) denotes the absolute value. The function is given by \[ |x | = \begin{cases} x & x \geq 0 \\ -x & x < 0 \\ \end{cases} \] For example, \( |3| = 3, |-2| = 2 \).

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