On parabola $y^2=2px$ at point $A$, a line $L_1$ passes that is tangent to the parabola and cuts $x$ axis at point $B$. From $A$, a line $L_2$ passes that is perpendicular to $x$ axis and cuts the parabola at point $C$.
A line $L_3$ passes point $B$ and is perpendicular to $x$ axis. A line $L_4$ passes point C and is parallel to $x$ axis.
Find the set of points $F$ where $L_3$ and $L_4$ intersect.
Book's answer is $y^2=-2px$
My intention was to present point $A$ as $(T,K)$ and eventually express $L_3$ and $L_4$ that way, find the expression of $T$ and $K$, and place them in the parabola's equation for the answer. Although I feel like the answer is there, I got lost. Would appreciate help