Let \( ABC \) and \( A_1B_1C_1 \) be two triangles such that the segments \( AA_1 \) and \( BC \) intersect, the segments \( BB_1 \) and \( AC \) intersect, and the segments \( CC_1 \) and \( AB \) intersect. If it is known that there exists a point \( X \) inside both triangles such that \[ \begin{aligned} \angle XAB &= \angle XA_1B_1, &\angle XBC &= \angle XC_1A_1, &\angle XCA &= \angle XB_1C_1,\\ \angle XAC &= \angle XB_1A_1, &\angle XBA &= \angle XA_1C_1, &\angle XCB &= \angle XC_1B_1. \end{aligned} \]Prove that the lines \( AC_1 \), \( BB_1 \), and \( CA_1 \) are concurrent or parallel.