
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 $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with a vertex at $(0,1)$ and a focal length of $2\sqrt{3}$. The left and right vertices of the ellipse $E$ are $A$, $B$, and $P$ is a moving point on the ellipse $E$ different from $A$ and $B$. The line $PB$ intersects the line $x=4$ at point $T$, and $AT$ intersects the ellipse $E$ at another point $Q$.  
$(1)$ Find the standard equation of the ellipse $E$;  
$(2)$ Determine whether the line $PQ$ passes through a fixed point on the $x$-axis. If it does, find the coordinates of that fixed point; if not, explain the reason.

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