
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 eccentricity $\frac{\sqrt{3}}{2}$, the points $F_1$ and $F_2$ are the left and right foci of the ellipse, respectively, and point $P$ is the intersection of the ellipse with the y-axis. The area of triangle $PF_1F_2$ is $\sqrt{3}$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ Given a line $l: kx+m$ intersects the ellipse $C$ at points $A$, $B$, and given $P(0,1)$, when the sum of the slopes of lines $PA$ and $PB$ is $2$, determine whether line $l$ passes through a fixed point. If it does, find the coordinates of the fixed point; if not, explain the reasons.

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