
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a < b > 0)$ with left focus $F$ and point $A(0,b)$, the distance from the origin $O$ to the line $FA$ is $\frac{\sqrt{2}}{2}b$.

1. Find the eccentricity $e$ of the ellipse $C$.
2. If the focus $F$ is symmetric about the line $l: 2x+y=0$ with point $P$ on the circle $O: x^{2}+y^{2}=4$, find the equation of the ellipse $C$ and the coordinates of point $P$.

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