
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 the ellipse C: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a>b>0)$$ with left and right foci $F_1$ and $F_2$, left vertex $A$, and eccentricity $\frac{\sqrt{2}}{2}$. Point $B$ is a moving point on the ellipse, and the maximum area of $\triangle ABF_1$ is $\frac{\sqrt{2}-1}{2}$.

1. Find the equation of ellipse C.
2. Let line $l$ pass through point $F_1$ and intersect ellipse C at two distinct points $M$ and $N$. The perpendicular bisector of segment $MN$ is $l'$. If line $l'$ intersects line $l$ at point $P$ and line $x=2$ at point $Q$, find the minimum value of $\frac{|PQ|}{|MN|}$.

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