
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the plane rectangular coordinate system $xOy$, given the ellipse $C$: $\frac{x^2}{4}+\frac{y^2}{3}=1$ with left and right foci $F_{1}$ and $F_{2}$, a line $l$ passing through point $T(4,0)$ (not coinciding with the $x$-axis) intersects the ellipse $C$ at points $A$ and $B$ (point $A$ lies between point $T$ and $B).$  
$(1)$ Let the slopes of lines $AF_{2}$ and $BF_{2}$ be $k_{1}$ and $k_{2}$, respectively. Find the value of $k_{1}+k_{2}$.  
$(2)$ Suppose the line $AF_{1}$ intersects $BF_{2}$ at point $M$. Find the value of $MF_{1}-MF_{2}$.

Present the answer in LaTex format: \boxed{Your answer}
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