
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 system $xoy$, it is known that the eccentricity of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $\frac{\sqrt{3}}{2}$, and its left and right foci are $F_1$ and $F_2$, respectively. A circle centered at $F_1$ with a radius of $3$ intersects with another circle centered at $F_2$ with a radius of $1$, and their intersection points lie on the ellipse $C$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ The line $y=k(x-1) (k \neq 0)$ intersects the ellipse $C$ at points $A$ and $B$, and point $M$ is the right vertex of the ellipse $C$. The lines $AM$ and $BM$ intersect the $y$-axis at points $P$ and $Q$, respectively. Determine whether the circle with diameter $PQ$ passes through a fixed point on the $x$-axis. If so, find the coordinates of the fixed point; if not, explain why.

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