The harmonic series $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$ is well known to be divergent.

If we however omit from this series every term where the denominator has a $9$ in it, the series remarkably enough converges to approximately $22.9206766193$.
This modified harmonic series is called the Kempner series.

Let us now consider another modified harmonic series by omitting from the harmonic series every term where the denominator has $3$ or more equal consecutive digits. One can verify that out of the first $1200$ terms of the harmonic series, only $20$ terms will be omitted.
These $20$ omitted terms are:

$$\frac{1}{111},\frac{1}{222},\frac{1}{333},\frac{1}{444},\frac{1}{555},\frac{1}{666},\frac{1}{777},\frac{1}{888},\frac{1}{999},\frac{1}{1000},\frac{1}{1110},$$ $$\frac{1}{1111},\frac{1}{1112},\frac{1}{1113},\frac{1}{1114},\frac{1}{1115},\frac{1}{1116},\frac{1}{1117},\frac{1}{1118},\frac{1}{1119}.$$

This series converges as well.

Find the value the series converges to.
Give your answer rounded to $10$ digits behind the decimal point.