
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
The ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) has its two foci and one of its vertices forming an isosceles right triangle. The chord formed by the intersection of the line $x+y=0$ with the circle centered at the right vertex of ellipse $C$ with radius $2b$ is of length $2\sqrt{3}$.
(Ⅰ) Find the standard equation of ellipse $C$;
(Ⅱ) Suppose a line $l$ passing through the right focus $F_2$ of ellipse $C$ intersects the ellipse at points $P$ and $Q$. If the parallelogram with $OP$ and $OQ$ as adjacent sides is a rectangle, find the equation of line $l$ that satisfies this condition.

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