HK-PC 2020

In $\Delta ABC$ , let $D$ be a point on side $BC$ . Suppose the incircle $\omega_1$ of $\Delta ABD$ touches sides $AB$ and $AD$ at $E,F$ respectively, and the incircle $\omega_2$ of $\Delta ACD$ touches sides $AD$ and $AC$ at $F,G$ respectively. Suppose the segment $EG$ intersects $\omega_1$ and $\omega_2$ again at $P$ and $Q$ respectively. Show that line $AD$ , tangent of $\omega_1$ at $P$ and tangent of $\omega_2$ at $Q$ are concurrent.

For a sequence with some ones and zeros, we count the number of continuous runs of equal digits in it. (For example the sequence $011001010$ has $7$ continuous runs: $0,11,00,1,0,1,0$ .) Find the sum of the number of all continuous runs for all possible sequences with $2019$ ones and $2019$ zeros.