Suppose that

\[ A = \sum_{i=0}^\infty \left( \frac{1}{ 2i+1} - \frac{1} { 2i + 2 } \right). \]

What is the value of

\[ \sum_{i=0}^\infty \left(\frac{ 1 } { 4i+1} + \frac{1}{ 4i+3} - \frac{1}{2i+2} \right)? \]

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