If \(z\) is a complex number such that
\[
\left|9z+\frac{1}{z}\right|=8,
\]
then the maximum value of \(|z|\) is
Show Hint
For expressions involving
\[
\left|az+\frac{b}{z}\right|,
\]
let
\[
|z|=r
\]
and use
\[
\boxed{
\left|a+b\right|
\ge
\Big||a|-|b|\Big|
}
\]
to obtain inequalities involving only \(r\).