
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 its 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 symmetric point P of point F about the line $l: 2x+y=0$ is 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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