
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 it passes through point $A(0,-1)$

(I) Find the equation of the ellipse

(II) If a line passing through point $(0,\frac{3}{5})$ intersects the ellipse at points $M$ and $N$ (where points $M$ and $N$ are not the same as point $A$),

(i) Find the value of $\overrightarrow{AM}\cdot \overrightarrow{AN}$; (ii) Find the equation of line $MN$ when $\Delta AMN$ is an isosceles right triangle.

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