That's a Circle, right?

Geometry Level 5

Above is the graph of the equation \[x^2+y^2-|2x+8|+2y=16\] If the area of the figure is denoted \(A\), then find \[\lfloor1000\{A\}\rfloor\]


Details and Assumptions
\(\{N\}\) denotes the fractional part of \(N\).
In other words, \(\{N\}=y\) where \(N=x+y\), \(x\) is an integer and \(0\le y < 1\).

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