
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 $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ (where $a>b>0$), its left and right vertices are A($-\sqrt {2}$, 0) and B($\sqrt {2}$, 0) respectively, and its eccentricity $e= \frac { \sqrt {2}}{2}$. A line perpendicular to the x-axis is drawn from any point P on the ellipse to point Q on the x-axis, and point C is on the extension line of QP such that $|PC|=(\sqrt {2}-1)|PQ|$.  
(1) Find the equation of the ellipse;  
(2) Find the equation of the trajectory E of the moving point C;  
(3) Suppose a line MN passes through the right focus of the ellipse and intersects the ellipse at points M and N, and $|MN|= \frac {8 \sqrt {2}}{7}$, find the equation of line MN.

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