
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 that point $P$ is on the parabola $x^{2}=2y$, a vertical line is drawn from point $P$ to the $x$-axis, with the foot of the perpendicular being $H$. Point $Q$ satisfies $\overrightarrow{PQ}=\frac{1}{2}\overrightarrow{PH}$.
(1) Find the equation of the trajectory $E$ of point $Q$;
(2) Point $M(-4,4)$, a line passing through point $N(4,5)$ with slope $k$ intersects trajectory $E$ at points $A$ and $B$. Let the slopes of lines $MA$ and $MB$ be $k_{1}$ and $k_{2}$, respectively. Find the minimum value of $|k_{1}-k_{2}|$.

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