Albert chooses a positive integer $k$, then two real numbers $a, b$ are randomly chosen in the interval $[0,1]$ with uniform distribution.
The square root of the sum $(k \cdot a + 1)^2 + (k \cdot b + 1)^2$ is then computed and rounded to the nearest integer. If the result is equal to $k$, he scores $k$ points; otherwise he scores nothing.
For example, if $k = 6$, $a = 0.2$ and $b = 0.85$, then $(k \cdot a + 1)^2 + (k \cdot b + 1)^2 = 42.05$.
The square root of $42.05$ is $6.484\cdots$ and when rounded to the nearest integer, it becomes $6$.
This is equal to $k$, so he scores $6$ points.
It can be shown that if he plays $10$ turns with $k = 1, k = 2, \dots, k = 10$, the expected value of his total score, rounded to five decimal places, is $10.20914$.
If he plays $10^5$ turns with $k = 1, k = 2, k = 3, \dots, k = 10^5$, what is the expected value of his total score, rounded to five decimal places?
solutions/s285.md.