
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 $\frac{\sqrt{6}}{3}$, a circle centered at $M(1,0)$ with the ellipse's minor axis as its radius is tangent to the line $x-y+\sqrt{2}-1=0$.

1. Find the standard equation of the ellipse $C$.
2. Given point $N(3,2)$, let line $l$ pass through point $M$ and intersect ellipse $C$ at points $A$ and $B$. If the slopes of lines $AN$ and $BN$ are $k\_1$ and $k\_2$, respectively, determine if $k\_1 + k\_2$ is a constant value. If it is, find the value; if not, explain why.

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