
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$ and a length of the minor axis of $4$. <br/>$(1)$ Find the equation of the ellipse; <br/>$(2)$ A chord passing through $P(2,1)$ divides $P$ in half. Find the equation of the line containing this chord and the length of the chord.

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