
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
A moving point P(x, y) has a ratio of distance to a fixed point F(1, 0) and to a fixed line x=2 as $\frac{\sqrt{2}}{2}$.  
(Ⅰ) Find the equation of the trajectory of P;  
(Ⅱ) A line l (not coinciding with the x-axis) passing through point F(1, 0) intersects the trajectory from (Ⅰ) at two points M and N. Investigate whether there exists a fixed point E(t, 0) on the x-axis such that any point on the x-axis (different from points E and F) has equal distances to lines EM and EN. If it exists, find the value of t; if not, explain why.

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