
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 an eccentricity of $\frac{\sqrt{3}}{2}$ and passes through the point $M(4, 1)$.
(I) Find the equation of ellipse $C$;
(II) If line $l$: $y = x + m\ (m \neq -3)$ intersects with ellipse $C$ at points $P$ and $Q$, and the slopes of lines $MP$ and $MQ$ are denoted as $k\_1$ and $k\_2$ respectively, investigate whether $k\_1 + k\_2$ is a constant value. If it is, find this constant value; if not, provide a reason.

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