
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 parabola $\Gamma: y^{2}=4\sqrt{3}x$ with focus $F_{1}$ coincides with one of the foci of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a > b > 0)$. The directrix of $\Gamma$ intersects the $x$-axis at $F_{1}$. If $\Gamma$ and $C$ intersect at points $A$ and $B$, and the sum of the distances from point $A$ to $F_{1}$ and $F_{2}$ is $4$.

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

(Ⅱ) If a line $l$ that does not pass through the origin and has a defined slope intersects the ellipse $C$ at points $G$ and $H$, and the area of $\triangle OGH$ is $1$, with the midpoint of segment $GH$ being $P$. Are there two fixed points $M$ and $N$ on the $x$-axis, symmetric about the origin, such that the product of the slopes of lines $PM$ and $PN$ is a constant? If so, find the coordinates of the two fixed points $M$, $N$ and the value of the constant; if not, explain why.

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