
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 $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ with eccentricity $$\frac { \sqrt {6}}{3}$$.
(I) If the distance from the origin to the line $x+y-b=0$ is $\sqrt {2}$, find the equation of the ellipse;
(II) Let a line $l$ that passes through the right focus of the ellipse and has a slope of 45° intersect the ellipse at points A and B.
(i) When $|AB|= \sqrt {3}$, find the value of $b$;
(ii) For any point M on the ellipse, if $\overrightarrow {OM}=\lambda \overrightarrow {OA}+\mu \overrightarrow {OB}$, find the relationship between the real numbers $\lambda$ and $\mu$.

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