
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
In the Cartesian coordinate plane, the two vertices of $\triangle ABC$ are $B(0,-1)$ and $C(0,1)$. Points $P$ and $Q$ in the plane simultaneously satisfy the following conditions:
1. $\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC} = \overrightarrow{0}$;
2. $|\overrightarrow{QA}|=|\overrightarrow{QB}|=|\overrightarrow{QC}|$;
3. $\overrightarrow{PQ} \parallel \overrightarrow{BC}$.
(1) Find the equation of the trajectory $E$ of vertex $A$;
(2) Draw two mutually perpendicular lines $l_{1}$ and $l_{2}$ passing through the point $F( \sqrt {2},0)$. The chords $A_{1}B_{1}$ and $A_{2}B_{2}$ formed by the intersection of lines $l_{1}$, $l_{2}$ and the trajectory $E$ of point $A$ have midpoints $M$ and $N$, respectively.
(i) Find the minimum value of the area $S$ of the quadrilateral $A_{1}A_{2}B_{1}B_{2}$;
(ii) Determine whether the line $MN$ always passes through a fixed point. If it does, find the fixed point; if not, explain why.

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