
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 the parametric equation of line $l$ as $\begin{cases} x= \frac{1}{2}+ \frac{ \sqrt{2}}{2}t \\ y= \frac{1}{2}- \frac{ \sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter), and the parametric equation of ellipse $C$ as $\begin{cases} x=2\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter). In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinates of point $A$ are $(2, \frac{\pi}{3})$.

$(1)$ Find the Cartesian coordinate equation of ellipse $C$ and the Cartesian coordinates of point $A$;

$(2)$ Line $l$ intersects ellipse $C$ at points $P$ and $Q$, find the area of $\triangle APQ$.

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