
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 one vertex at A(2,0), and an eccentricity of $$\frac { \sqrt {2}}{2}$$, the line $y=k(x-1)$ intersects the ellipse $C$ at two distinct points $M$ and $N$.  
(1) Find the equation of the ellipse $C$;  
(2) When the area of $\triangle AMN$ is $$\frac {4 \sqrt {2}}{5}$$, find the value of $k$.

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