
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 that the eccentricity of the ellipse $\frac{{y}^{2}}{{a}^{2}}+\frac{{x}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ is $\frac{\sqrt{3}}{2}$, and the length of the minor axis is $4$. The ellipse intersects the line $y=x+2$ at points $A$ and $B$. <br/>$(1)$ Find the equation of the ellipse;<br/>$(2)$ Find the length of the chord $|AB|$.

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