
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, two vertices of triangle $\vartriangle ABC$ 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 point $F(\sqrt{2}, 0)$. The chords $A_1B_1$ and $A_2B_2$ formed by the intersection of lines $l_1$, $l_2$ and trajectory $E$ have midpoints $M$ and $N$ respectively. Determine whether line $MN$ always passes through a fixed point. If it does, find the fixed point; if not, explain the reason.

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