
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ passes through the point $M(1, \frac{3}{2})$, and its eccentricity is $\frac{1}{2}$.
(I) Find the equation of the ellipse $C$;
(II) Suppose the line $l: y=kx+m \; (|k|\leqslant \frac{1}{2})$ intersects the ellipse $C$ at points $A$ and $B$. Construct a parallelogram $OAPB$ with sides $OA$ and $OB$, where $P$ is a point on the ellipse $C$, and $O$ is the origin. Find the range of values for $|OP|$.

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