
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
Given a point $M(x,y)$ and a fixed point $F_{2}(1,0)$, the ratio of the distance from $M$ to $F_{2}$ and its distance to the line $x=4$ is a constant $\frac{1}{2}$  
$(1)$ Find the equation of the trajectory of point $M$ and explain what shape the trajectory is;  
$(2)$ If the coordinates of point $F_{1}$ are $(-1,0)$, and a line $l$ passing through $F_{2}$ intersects the trajectory of point $M$ at two distinct points $A$ and $B$, find the maximum value of the area of $\triangle F_{1}AB$.

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