A Min Focal Sum on a Chord

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has left and right foci $F_1$, $F_2$; the point $M\left(1,\dfrac 32\right)$ lies on $C$, and in $\triangle F_1MF_2$, $\tan\angle MF_1F_2=\dfrac 34$.

1. Find the equation of the ellipse $C$. 2. A line $l$ through $P(1,3)$ meets $C$ at two points $A$ and $B$ (with $A$ above $B$). A point $Q$ on the segment $AB$ satisfies $\dfrac{|PA|}{|PB|}=\dfrac{|QA|}{|QB|}$. Find the minimum value of $|QF_1|+|QF_2|$.