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

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