In a right triangle $ABC$ (where $\angle C = 90^\circ$), it is given that $AC = 4$ cm and $BC = 3$ cm. The points $A_1$, $B_1$, and $C_1$ are positioned such that $AA_1 \parallel BC$, $BB_1 \parallel A_1C$, and $CC_1 \parallel A_1B_1$. Additionally, $\angle A_1B_1C_1 = 90^\circ$ and $A_1B_1 = 1$ cm. Find the length of $B_1C_1$.

Let's think step by step and output the final answer within \boxed{}.
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