
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 line m passing through point A (-4, 0) intersects with the parabola G: $x^2=2py$ ($p>0$) at points B and C.  
(1) When the slope of line m is $\frac{1}{2}$, $\overrightarrow{AC}=4\overrightarrow{AB}$, find the equation of the parabola G;  
(2) Let M be the midpoint of B and C, using the parabola G found in (1), try to find the equation of the trajectory of point M.

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