
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 a major axis length of 6, and the common chord length between ellipse C and circle $M:(x-2)^{2}+y^{2}= \frac{40}{9}$ is $\frac{4\sqrt{10}}{3}$.

1. Find the equation of ellipse C.
2. A line $l$ passing through point P(0, 2) with a slope of $k (k>0)$ intersects ellipse C at two points A and B. Determine if there exists a point D on the x-axis such that ΔADB is an isosceles triangle with base AB. If it exists, find the range of the x-coordinate values for point D. If not, explain the reason.

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