
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: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$  ($a > b > 0$) with an eccentricity of $$\frac{1}{2}$$. The left and right foci are F<sub>1</sub> and F<sub>2</sub>, respectively. A circle centered at F<sub>1</sub> with radius 3 intersects with another circle centered at F<sub>2</sub> with radius 1, and their intersection points lie on the ellipse C.  
(Ⅰ) Find the equation of ellipse C.  
(Ⅱ) Segment PQ is a chord of ellipse C passing through point F<sub>2</sub>, and $$\overrightarrow{PF_2} = \lambda \overrightarrow{F_2Q}$$.
(i) Find the perimeter of triangle PF<sub>1</sub>Q.  
(ii) Find the maximum area of the inscribed circle of triangle PF<sub>1</sub>Q, and determine the value of the real number λ for which this maximum is attained.

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