Maclaurin 2026 — Square corner removed (perimeter and area)

A square $ABCD$ has side length $1$. Point $X$ lies on $AB$ and $Y$ lies on $AD$ with $AX = x$, $AY = y$ and $x < y$. Triangle $AXY$ is removed to leave pentagon $BCDYX$.

The perimeter of $BCDYX$ is $\frac{29}{30}$ of the perimeter of $ABCD$, and the area of $BCDYX$ is also $\frac{29}{30}$ of the area of $ABCD$.

Find the length of $AX$.