
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 points $A(-2,0)$ and $P(1, \frac{3}{2})$ on the ellipse $M: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$, and two lines with slopes $k$ and $-k (k>0)$ passing through point $P$ intersect ellipse $M$ at points $B$ and $C$.
(I) Find the equation of ellipse $M$ and its eccentricity.
(II) If quadrilateral $PABC$ is a parallelogram, find the value of $k$.

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