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

1. Find the standard equation of the ellipse $C$.
2. Given points $N(3,2)$ and $P(m,n) (m \neq 3)$, draw a line $l$ through point $M$ that intersects the ellipse $C$ at points $A$ and $B$. If the slopes of the lines $AN$, $NP$, and $BN$ are $k_{1}$, $k_{2}$, and $k_{3}$, respectively, and $k_{1}+k_{3}=2k_{2}$, find the relationship between $m$ and $n$.

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