
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 an 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 with radius equal to the length of the minor axis of the ellipse is centered at $M(1,0)$, and is tangent to the line $x - y + \sqrt{2} - 1 = 0$.

1. Find the standard equation of the ellipse $C$.
2. Given a point $N(3,2)$ and another point $P(m,n)$ ($m \neq 3$) in the plane, draw a line $l$ passing through point $M$ that intersects the ellipse $C$ at points $A$ and $B$. Let $k_1$, $k_2$, and $k_3$ be the slopes of the lines $AN$, $NP$, and $BN$, respectively. If $k_1 + k_3 = 3k_2$, find the relation between $m$ and $n$.

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