Let $\Gamma_1$ and $\Gamma_2$ be the circles with diameters $AP$ and $AQ$. Let $T$ be another point of intersection of the circles $\Gamma_1$ and $\Gamma_2$. Let $Q_1$ be another point of intersection of the circle $\Gamma_1$ and the line $AQ$, and $P_1$ the other point of intersection of the circle $\Gamma_2$ and the line $AP$. The circle $\Gamma_3$ passes through the points $T$, $P$ and $P_1$ and the circle $\Gamma_4$ passes through the points $T$, $Q$ and $Q_1$. Prove that the line containing the common chord of the circles $\Gamma_3$ and $\Gamma_4$ passes through$A$.