
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 the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with its left vertex at M and upper vertex at N. The slope of the line MN is $\frac{1}{2}$.
(I) Find the eccentricity of the ellipse.
(II) The line $l: y = \frac{1}{2}x + m$ ($m \neq 0$) intersects the ellipse at points A and C, and intersects the y-axis at point P. A square ABCD is constructed with AC as its diagonal. If $|BP| = \frac{\sqrt{10}}{2}$.
(i) Find the equation of the ellipse.
(ii) If point E is on line MN and satisfies $\angle EAC = 90^\circ$, find the equation of line AC when $|EC|$ is the longest.

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