--description--

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 ${(ka + 1)}^2 + {(kb + 1)}^2$ 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 ${(ka + 1)}^2 + {(kb + 1)}^2 = 42.05$. The square root of 42.05 is 6.484... 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 $k = 1, k = 2, \ldots, k = 10$, the expected value of his total score, rounded to five decimal places, is 10.20914.

If he plays ${10}^5$ turns with $k = 1, k = 2, k = 3, \ldots, k = {10}^5$, what is the expected value of his total score, rounded to five decimal places?

--hints--

pythagoreanOdds() should return 157055.80999.

assert.strictEqual(pythagoreanOdds(), 157055.80999);

--seed--

--seed-contents--

function pythagoreanOdds() {

  return true;
}

pythagoreanOdds();

--solutions--

// solution required