I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following:

Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, $L$ be a line on the plane. Let $L$ meets $A_1A_2$, $A_2A_3$, $A_3A_4$, $A_4A_5$, $A_5A_6$, $A_6A_1$ at $B_2$, $B_3$, $B_4$, $B_5$, $B_6$, $B_1$ respectively.

Let $C_1$ be a point on the line $A_1A_4$. Let $C_1B_2$ meets $A_2A_5$ at $C_2$. Let $C_2B_3$ meets $A_3A_6$ at $C_3$. Let $C_3B_4$ meets $A_1A_4$ at $C_4$. Let $C_4B_5$ meets $B_2B_5$ at $C_5$. Let $C_5B_6$ meets $A_3A_6$ at $C_6$. Let $C_6B_1$ meets $A_1A_4$ at $C_7$. Then:

1. Six points $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ lie on a conic if only if six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ lie on a conic.

2. $C_7 \equiv C_1$ if only if six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ lie on a conic.

3. if $C_7 \equiv C_1$ then the Pascal line of hexagon $A_1A_2A_3A_4A_5A_6$ and $C_1C_2C_3C_4C_5C_6$ and $L$ are concurrent.

Remark: When the conic through $A_1, A_2, A_3, A_4, A_5, A_6$ is the circumcircle and $L$ at infinity, the item 1 is the Tucker circle theorem. When the conic through $A_1, A_2, A_3, A_4, A_5, A_6$ is the Steiner inellipse and $L$ at infinity, item 2 is the Thomsen theorem.

My question: can you give a proof for the problem?