
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\ (a > b > 0)$ whose major axis is $\sqrt {2}$ times its minor axis. The line $y=-x+1$ intersects the ellipse $C$ at points $A$ and $B$, with the chord $AB$ having a length of $\frac{4\sqrt{5}}{3}$. Determine the equation of this ellipse.

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