Divisibility Comparison Between Factorials

Let $f_{5} (n)$ be the largest integer $x$ for which $5^{x}$ divides $n$ .
For example, $f_{5} (625000) = 7$ .

Let $T_{5} (n)$ be the number of integers $i$ which satisfy $f_{5} ((2 \cdot i - 1)!) < 2 \cdot f_{5} (i!)$ and $1 \leq i \leq n$ .
It can be verified that $T_{5} (10^{3}) = 68$ and $T_{5} (10^{9}) = 2408210$ .

Find $T_{5} (10^{18})$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.