# Can linear equations be hard?

Algebra Level 5

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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