
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 that the two foci of ellipse C are F1(-2$\sqrt {2}$,0) and F2(2$\sqrt {2}$,0), and its major axis length is 6.
1. Find the equation of ellipse C.
2. Let point H be (0,1). If the line y=kx+3 intersects ellipse C at two points M and N, and the sum of the slopes of HM and HN is 1, find the value of the real number k.

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