
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 an ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \left( a > b > 0 \right)$ with eccentricity $\dfrac{1}{2}$, and it passes through the point $\left( 1,\dfrac{3}{2} \right)$. If point $M\left( x_{0},y_{0} \right)$ is on the ellipse $C$, then the point $N\left( \dfrac{x_{0}}{a},\dfrac{y_{0}}{b} \right)$ is called an "elliptic point" of point $M$.

$(1)$ Find the standard equation of the ellipse $C$;

$(2)$ If the line $l:y=kx+m$ intersects the ellipse $C$ at points $A$ and $B$, and the "elliptic points" of $A$ and $B$ are $P$ and $Q$ respectively, and the circle with diameter $PQ$ passes through the origin, determine whether the area of $\Delta AOB$ is a constant value? If it is a constant, find the value; if not, explain why.

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