
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 $$C_{1}: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ whose left and right foci are F<sub>1</sub> and F<sub>2</sub>, respectively, and it intersects with the line x+y-1=0 at points A and B.  
(1) If the two foci of the ellipse C<sub>1</sub> are the vertices of the hyperbola $$C_{2}: x^{2}- \frac {y^{2}}{2}=1$$, and the perimeter of the triangle △PF<sub>1</sub>F<sub>2</sub> formed by any point P on the ellipse and the left and right foci F<sub>1</sub>, F<sub>2</sub> is $$2 \sqrt {3}+2$$, find the standard equation of the ellipse C<sub>1</sub>;  
(2) Under the condition of (1), find the length of chord AB;  
(3) When the eccentricity e of the ellipse satisfies $$\frac { \sqrt {3}}{3}≤e≤ \frac { \sqrt {2}}{2}$$, and the circle with AB as the diameter passes through the origin O, find the range of the length of the major axis of the ellipse.

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