
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 Г: $$\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity is $$\frac { \sqrt {2}}{2}$$. The shortest line segment from $F_2$ to a point on the ellipse is $\sqrt {2} - 1$.  
(1) Find the standard equation of ellipse Г.  
(2) It is known that there exists a point P on Г such that the lines $PF_1$ and $PF_2$ intersect the ellipse Г at points A and B respectively. If $$\overrightarrow {PF_{1}} = 2 \overrightarrow {F_{1}A}$$ and $$\overrightarrow {PF_{2}} = \lambda \overrightarrow {F_{2}B}$$ ($\lambda > 0$), find the value of $\lambda$.

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