
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 $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1({a＞0,b＞0})$ passing through the point $D({1,\frac{{\sqrt{2}}}{2}})$ with eccentricity $e=\frac{{\sqrt{2}}}{2}$, where the left and right foci are $F_{1}$ and $F_{2}$ respectively. Points $P$ and $Q$ are two points on the ellipse $C$ above the $x$-axis.<br/>$(1)$ If $PF_{1}$<span class="special-symbol">∥</span>$QF_{2}$ and $|PF_{1}|+|QF_{2}|=2$, find the equation of the line $QF_{2}$;<br/>$(2)$ Extend $PF_{1}$ and $PF_{2}$ to intersect the ellipse $C$ at points $M$ and $N$ respectively. Let $\overrightarrow{M{F_1}}=λ\overrightarrow{{F_1}P}$ and $\overrightarrow{N{F_2}}=n\overrightarrow{{F_2}P}$. Find the minimum value of $\lambda n$.

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