KöMaL 2007 B3964

The lengths of the legs in a right-angled triangle ABC are $AC$=3 and $BC$=4. The point $A$is shifted in a direction parallel to $BC$ to the point $A_1$, then the point $B$ is shifted in a direction parallel to the line $A_1 C$ to the point $B_1$, and finally $C$ is shifted in a direction parallel to $A_1 B_1$ to the point $C_1$. The resulting triangle $A_1 B_1 C_1$ is right-angled at $B_1$ and its leg $A_1 B_1$ is 1 unit long. How long is the leg $B_1 C_1$?