
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
In the rectangular coordinate system $xoy$, the ratio of the distance from the moving point $M$ to the point $F(1,0)$ and its distance to the line $x=2$ is $\frac{\sqrt{2}}{2}$.
(I) Find the equation of the trajectory $E$ of the moving point $M$.
(II) Suppose the line $y=kx+m (m\neq 0)$ intersects the curve $E$ at points $A$ and $B$, and intersects the $x$-axis and $y$-axis at points $C$ and $D$ respectively (where $C$ and $D$ are between $A$ and $B$ or both outside of $A$ and $B$). Determine whether there exists a fixed value $k$ such that for any real number $m$ satisfying the conditions, the area of triangle $OAC$ is equal to the area of triangle $OBD$. If it exists, find the value of $k$; if not, explain the reason.

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