
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
The eccentricity of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left(a > b > 0\right)$ is $\frac{\sqrt{6}}{3}$, and the distance from one endpoint of the minor axis to the right focus is $\sqrt{3}$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Suppose the line $y=x+1$ intersects the ellipse $C$ at points $A$ and $B$. Find the distance between points $A$ and $B$.

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