
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 a line $l$ intersects with the ellipse $\frac {y^{2}}{a^{2}}+ \frac {x^{2}}{b^{2}}=1 (a>b>0)$ at two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$. The maximum and minimum distances from a point on the ellipse to the lower focus are $2+ \sqrt {3}$ and $2- \sqrt {3}$, respectively. Let vector $\overrightarrow {m}=(ax_{1}, by_{1})$, $\overrightarrow {n}=(ax_{2}, by_{2})$, and $\overrightarrow {m} \perp \overrightarrow {n}$. Let $O$ be the origin.  
(Ⅰ) Find the equation of the ellipse;  
(Ⅱ) Determine whether the area of $\triangle AOB$ is a constant value. If yes, provide a proof; if not, explain why.

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