
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
Let ellipse $G$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ intersect the positive $y$-axis at point $B$ and have the right focus at point $F$. Given that points $B$ and $F$ lie on circle $C$: $x^{2} + y^{2} - 2x - 2y = 0$.
- $(1)$ Find the equation of ellipse $G$.
- $(2)$ If line $l$ passes through point $C$ and intersects ellipse $G$ at points $M$ and $N$, with $C$ being the midpoint of segment $MN$, find $|MN|$.

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