
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 an ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $e$, a line $l$ passing through the right focus $F$ intersects $C$ at points $A$ and $B$. When the slope of $l$ is 1, the distance from the origin $O$ to $l$ is $\frac{\sqrt{2}}{2}$.  
(Ⅰ) Find the values of $a$ and $b$;  
(Ⅱ) Determine whether there exists a point $P$ on $C$ such that when $l$ rotates around $F$ to a certain position, the equation $OP = OA + OB$ holds. If such a point exists, find all coordinates of $P$ and the equation of $l$; if not, explain why.

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