
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 a parabola with its vertex at the origin and its focus F on the x-axis, and it passes through the point (4, 4).  
(Ⅰ) Find the standard equation of the parabola and the coordinates of the focus;  
(Ⅱ) Let point P be a moving point on the parabola, and point M be the midpoint of PF. Find the equation of the trajectory of point M.

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