
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 lines $l_1: y = \frac{\sqrt{2}}{2}x$ and $l_2: y = -\frac{\sqrt{2}}{2}x$, point A moves on $l_1$ and point B moves on $l_2$ such that the length of segment AB is constantly $2\sqrt{2}$.
(I) Find the trajectory equation of the midpoint C of segment AB;
(II) Suppose line $l_1: y = kx + m$ intersects the trajectory of point C at points M and N, and O is the origin. If $\vec{k_{OM}} \cdot \vec{k_{ON}} = \frac{5}{4}$, find the range of the distance from the origin O to line $l$.

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