A Very Special Right Triangle

Right triangle △ $ABC$ (right angle $CBA$ ) is very special:

$\overline{AD}$ bisects angle $BAC$ .

$\overline{BE}$ is perpendicular to $\overline{AC}$ .

$\overline{CF}$ bisects $\overline{AB}$ into $\overline{AF}=\overline{BF}=1$ .

And most importantly, $\overline{AD}$ , $\overline{BE}$ , and $\overline{CF}$ all intersect each other at the same exact point $P$ .

Diagram (*not to scale):

Given that $\overline{CB}$ lies along the $x$ -axis, and that point $B$ is at the origin, what is the absolute value of the $x$ -coordinate of $E$ , to the nearest hundred-thousandeth?