Given that

\[f\left( x \right) =\left| \left| x \right| -1 \right| \] \[{ P }_{ 0 }(x)=f(x)\] \[{ P }_{ n+1 }(x)={ f(P }_{ n }(x))\]

\[A_{k}=\left| \lim _{ n\rightarrow \infty }{ \int _{ 0 }^{ k }{ { P }_{ n }(x) } -0.5dx } \right| \]

\[A=\max{A_{k}}\]

And that \(k_{0}\) is the minimum value such that \(A=A_{k_{0}}\), where \(k\) ranges over all the positive reals \(k>0\).

Find \[\left\lfloor 1000(A+k_{0}) \right\rfloor \]

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