
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 its left focus at $F_1(-1, 0)$, and the tangent line at the point ($\sqrt{2}, 1$) on the parabola $x^2 = 2py$ passes through the bottom vertex of ellipse C.

(1) Find the standard equation of ellipse C;  
(2) A moving line $l$ passing through point $F_1$ intersects ellipse C at points A and B (different from the endpoints of the major axis). Is there a real constant $\lambda$ such that $|\overrightarrow{F_{2}A} - \overrightarrow{F_{2}B}| = \lambda \overrightarrow{F_{1}A} \cdot \overrightarrow{F_{2}B}$ always holds? If it exists, find the value of $\lambda$; if not, explain why;  
(3) Under the condition of (2), find the range of the area of the inscribed circle in $\triangle ABF_{2}$ ($F_{2}$ is the right focus of ellipse C).

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