
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 Cartesian coordinate plane $(xOy)$, the ratio of the distance from the moving point $S$ 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 $S$;
(II) Draw a line $l$ passing through point $F$ and not perpendicular to the $x$-axis, intersecting the trajectory $E$ at points $P$ and $Q$. Does there exist a point $M(m,0)$ on the segment $OF$ such that $( \overrightarrow{MP}+ \overrightarrow{MQ})\cdot \overrightarrow{PQ}=0$? If so, find the range of $m$; if not, explain the reason.

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