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Evaluate the sum of all real values of xxx that satisfy the following equation.
8x+1+44⋅2x+1+64=19⋅4x+1 8^{\sqrt{x+1}} + 44 \cdot 2^{\sqrt{x+1}} + 64 = 19 \cdot 4^{\sqrt{x+1}} 8x+1+44⋅2x+1+64=19⋅4x+1
This problem was adapted from ITA's 2013 Math Paper.
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