
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 the parabola $C: x^2 = 2py$ ($p > 0$) with focus $F$, a tangent line $l$ to the parabola at point $P$ intersects the x-axis at point $D$ and the y-axis at point $Q$. When $|FD| = 2$, $\angle PFD = 60^\circ$.

(1) Determine the shape of $\triangle PFQ$ and find the equation of parabola $C$;  
(2) If points $A$ and $B$ are on parabola $C$, and satisfy $\overrightarrow{AM} + \overrightarrow{BM} = 0$, where point $M(2, 2)$, and there exists a point $H$ on parabola $C$, different from $A$ and $B$, such that the circle passing through points $A$, $B$, and $H$ has the same tangent at point $H$ as the parabola, find the coordinates of point $H$.

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