
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
A moving point $P$ has a constant sum of distances to two fixed points $F_1(-\sqrt{3},0)$ and $F_2(\sqrt{3},0)$ equal to $4$;

$(1)$ Find the equation of the trajectory $B$ of the moving point $P$;

$(2)$ Given that point $G$ is any point on the curve $B$, $GH$ is perpendicular to the $x$-axis at point $H$, and point $G$ is the midpoint of the segment $RH$, find the equation of the trajectory $D$ of point $R$;

$(3)$ The curve $D$ is translated upwards by $3$ units to obtain curve $C$. A moving line $l$ passing through point $A(-1,0)$ intersects curve $C$ at points $P$ and $Q$, with $M$ being the midpoint of $PQ$. The line $l$ intersects line $m$: $x+3y+6=0$ at point $N$. Determine whether $\vec{AM} \cdot \vec{AN}$ is related to the slope of line $l$. If it is not related, find its value; if it is related, explain why.

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