
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 left and right foci $F_{1}$ and $F_{2}$, respectively, left vertex $A$, and upper vertex $B(0,1)$. The area of $\triangle AB{F_{1}}$ is $\frac{\sqrt{2}-1}{2}$.

(1) Find the equation of ellipse $C$.
(2) Let line $l$: $y=k(x+1)$ intersect with ellipse $C$ at two distinct points $M$ and $N$. $P$ is the midpoint of segment $MN$. If line $m$ passing through point $F_{2}$ is perpendicular to line $l$ at point $Q$, find the range of values for $|PQ| \cdot |F_{1}Q|$.

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