
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
The vertex of the parabola is at the origin, the focus is on the x-axis, and the point (4, 4) lies on the parabola, with the focus denoted as F;
(1) Determine the coordinates of the focus and the standard equation of the parabola;
(2) P is a moving point on the parabola, M is the midpoint of PF, find the equation of the trajectory of M.

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