
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 equations of a line $l$: 
$$
\begin{cases}
x= \frac {1}{2}+ \frac { \sqrt {2}}{2}t \\
y= \frac {1}{2}- \frac { \sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
The ellipse $C$ has parametric equations:
$$
\begin{cases}
x=2\cos \alpha \\
y=\sin \alpha
\end{cases}
(\alpha \text{ is the parameter}).
$$
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 $Q$. Calculate the area of triangle $APQ$.

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