
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 $T$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, a line passing through the left focus $F$ intersects the ellipse at points $A$ and $B$. The midpoint of segment $AB$ is $M(- \frac{2}{3}, \frac{1}{3})$.
(1) Find the equation of the ellipse;
(2) A line $l$ passing through the right focus intersects the circle $x^{2}+y^{2}=2$ at points $C$ and $D$, and intersects the ellipse $T$ at points $E$ and $G$. Given $|CD|= \sqrt{5}$, find $|EG|$.

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