
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 two points $M$ and $N$ moving on the lines $y=mx$ and $y=-mx (m > 0)$ respectively, with $|MN|=2$. Point $P$ satisfies $2 \overrightarrow{OP}= \overrightarrow{OM}+ \overrightarrow{ON} (O$ is the coordinate origin$)$, and the trajectory of point $P$ is denoted as curve $C$.
(I) Find the equation of curve $C$ and discuss the type of curve $C$;
(II) A line $l$ passing through point $(0,1)$ intersects curve $C$ at two different points $A$ and $B$. If for any $m > 1$, $\angle AOB$ is acute, find the range of the slope $k$ of line $l$.

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