
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 $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$ with a focal distance of $2\sqrt{6}$, and the sum of the distances from any point on the ellipse $C$ to the two foci of the ellipse is $6$.

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

$(2)$ Let the line $l:y=x+1$ intersect the ellipse $C$ at points $A$ and $B$, find the length of the chord $|AB|$.

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