
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 the hyperbola $C$ has the same asymptotes as the hyperbola $\frac{{y^2}}{4} - x^2 = 1$, and the distance from the upper focus of hyperbola $C$ to one of the asymptotes is $2$.  
$(1)$ Given $M(0,t)$ where $t > 4$, $N$ is any point on $C$, find the minimum value of $|MN|$.  
$(2)$ Given the moving line $l: y = kx + m$ where $k \neq \pm 2$ intersects the curve $C$ at exactly one point $P$, and the line $l_1$ passing through point $P$ perpendicular to $l$ intersects the two coordinate axes at $A(x_0, 0)$ and $B(0, y_0)$. Let $Q(x_0, y_0)$.  
$(i)$ Find the equation of the trajectory of point $Q$.  
$(ii)$ If in general, the equation of curve $C$ is $\frac{{y^2}}{{a^2}} - \frac{{x^2}}{{b^2}} = 1$, and the equation of the moving line $l$ is $y = kx + m$ where $k \neq \pm \frac{a}{b}$, write directly the equation of the trajectory of point $Q(x_0, y_0)$.

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