
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 $\frac{{x}^{2}}{{a}^{2}}$+$\frac{{y}^{2}}{{b}^{2}}$=1 (a>b>0) with a vertex at A(0,-3), a right focus at F, and |OA|=|OF|, where O is the origin. <br /> (Ⅰ) Find the equation of the ellipse; <br /> (Ⅱ) Given a point C such that 3$\overrightarrow{OC}$=$\overrightarrow{OF}$, point B lies on the ellipse (B is different from the vertex of the ellipse), the line AB is tangent to the circle with center at C at point P, and P is the midpoint of segment AB. Find the equation of the line AB.

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