
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 hyperbola equation $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a>0,b>0)$$, an ellipse C has its vertices at the foci of this hyperbola, and its foci at the vertices of the hyperbola.  
(1) When $$a= \sqrt {3}$$ and $b=1$, find the equation of ellipse C;  
(2) Under the condition of (1), the line $l: y=kx+ \frac {1}{2}$ intersects the y-axis at point P and intersects the ellipse at points A and B. If O is the origin, and the area ratio of $\triangle AOP$ to $\triangle BOP$ is 2:1, find the equation of line $l$;  
(3) If $a=1$, the ellipse C and the line $l': y=x+5$ have common points, find the minimum length of the major axis of this ellipse.

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