
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 a focal distance of $2\sqrt{3}$, and $C$ intersects the $y$-axis at points $A(0,-1)$ and $B(0,1)$.

$(1)$ Find the standard equation of the ellipse $C$ and its eccentricity;

$(2)$ Suppose point $P$ is a moving point on ellipse $C$ and is on the right side of the $y$-axis. The lines $PA$ and $PB$ intersect the line $x=3$ at points $M$ and $N$, respectively. If a circle with diameter $MN$ intersects the $x$-axis at points $E$ and $F$, find the range of the $x$-coordinate of point $P$ and the maximum value of $|EF|$.

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