
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 Cartesian coordinate plane $(xOy)$, the ratio of the distance from a moving point $P$ to a fixed point $F(-1,0)$ to the distance from $P$ to the fixed line $x=-4$ is $\frac{1}{2}$.
(1) Find the equation of the trajectory $C$ of the moving point $P$;
(2) If the minimum value of the distance from a moving point $N$ on the trajectory $C$ to a fixed point $M(m,0) (0 < m < 2)$ is $1$, find the value of $m$;
(3) Let points $A$ and $B$ be two moving points on the trajectory $C$, and let $A_1$ and $B_1$ be the other intersection points of lines $OA$ and $OB$ with the trajectory $C$, respectively. If the product of the slopes of lines $OA$ and $OB$ is $-\frac{3}{4}$, is the area $S$ of the quadrilateral $ABA_1B_1$ a fixed value? Please explain your reasoning.

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