
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 ellipse $C_{1}： \frac {x^{2}}{4}+ \frac {y^{2}}{b^{2}}=1$ and the curve $C_{2}$ which is symmetric to $C_{1}$ about the line $y=-x$. The points of intersection of $C_{1}$ and $C_{2}$ in the first, second, third, and fourth quadrants are $P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$ respectively. If the area of the quadrilateral $P_{1}P_{2}P_{3}P_{4}$ is 4, find the coordinates of point $P_{1}$ and the eccentricity of $C_{1}$.

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