
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 $M: \dfrac{x^{2}}{9}+\dfrac{y^{2}}{b^{2}}=1\left(b \gt 0\right)$ with one focus at $\left(2,0\right)$, let ellipse $N$ have its focus at the vertex on the minor axis of ellipse $M$, and ellipse $N$ passes through the point $(\dfrac{\sqrt{2}}{2}$,$\sqrt{3})$.
1. Find the equation of $N$.
2. If the line $y=x-2$ intersects ellipse $N$ at points $A$ and $B$, find $|AB|$.

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