
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 the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), with foci $F_1$ and $F_2$ respectively, and the symmetric point $M$ of $F_2$ with respect to the line $x-y+a=0$ lies on the line $3x+2y=0$.
$(1)$ Calculate the eccentricity of the ellipse;
$(2)$ If a line perpendicular to the x-axis passing through focus $F_2$ intersects the ellipse at a chord of length $3$, and a line $l$ with slope $\frac{1}{2}$ intersects the ellipse at points $A$ and $B$, determine whether there is a fixed point $P$ such that the sum of the slopes of $PA$ and $PB$ is a constant value. If such a point exists, find the coordinates of all the points $P$ that satisfy the condition; if not, explain why.

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