Cayley 2026 — Rhombus and parallelogram (prove rectangle)

$BCDF$ is a rhombus with side length $x$. Point $E$ lies on $DF$ and $ABCE$ is a parallelogram. Side $AB$ has length $y$ and $AF$ has length $z$. The perimeter of trapezium $ABCD$ is $10\%$ more than the perimeter of $BCDF$. The perimeter of $ABCE$ is $10\%$ less than the perimeter of $BCDF$.

Prove that $ABCE$ is, in fact, a rectangle.