
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
Consider an ellipse $C$ with the left and right foci denoted as $F_1$ and $F_2$, respectively, and defined by the equation
$$
\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \quad (a > b > 0).
$$
The ellipse passes through the point $Q(0, \sqrt{3})$. Denote $P$ as a point on the ellipse, and let $G$ be the centroid and $I$ be the incenter of triangle $\triangle PF_1F_2$. It is given that $IG$ is parallel to the line connecting $F_1$ and $F_2$.
1. Determine the equation of ellipse $C$.
2. Let $M$ be a point on the line $x-y=4$. Form two tangents to the ellipse $C$ from point $M$, denoted as $MA$ and $MB$, with $A$ and $B$ being the points of tangency. Determine whether the line $AB$ passes through a fixed point. If so, find the coordinates of this fixed point; if not, provide a reason.

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