
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 left and right foci $F_{1}$ and $F_{2}$, respectively, and an eccentricity of $\frac{\sqrt{2}}{2}$. The length of the shortest line segment connecting the midpoint of the line segment $F_{2}P$ (where $P$ is a point on the ellipse) to the ellipse is $\sqrt{2}-1$.

1. Find the standard equation of the ellipse $(Г)$.
2. If a point $P$ exists on the ellipse $(Г)$ such that the lines $PF_{1}$ and $PF_{2}$ intersect the ellipse $(Г)$ at points $A$ and $B$, respectively, and $\overrightarrow{PF_{1}}=2\overrightarrow{F_{1}A}$ 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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