
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
An ellipse $C$ with left and right foci $F\_1$ and $F\_2$ respectively, is given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$. The ellipse passes through point $Q(0, \sqrt{3})$. $P$ is a point on the ellipse, $G$ is the centroid of $\triangle PF\_1F\_2$, and $I$ is the incenter of $\triangle PF\_1F\_2$ with $IG$ parallel to $F\_1F\_2$.

1. Find the equation of ellipse $C$.
2. $M$ is a point on the line $x - y = 4$. Two tangent lines, $MA$ and $MB$, are drawn from point $M$ to the ellipse $C$, with $A$ and $B$ as the points of tangency. Determine whether the line $AB$ passes through a fixed point. If it does, find the coordinates of the fixed point. If not, provide a reason.

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