
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 its right focus at $F( \sqrt {2}，0)$, the length of the chord formed by the intersection of the ellipse and the line passing through point F perpendicular to the x-axis is 2.

1. Find the equation of ellipse C.
2. Let A and B be two moving points on ellipse C, and M be the midpoint of line segment AB. The slopes of lines AB and OM (where O is the coordinate origin) both exist and are denoted as $k_{1}$ and $k_{2}$, respectively. Determine whether the value of $k_{1}k_{2}$ is constant. If it is, find this constant value; if not, explain the reason.

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