
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
Definition: In the Cartesian coordinate system, if there are at least two distinct points on the graph of a function that are symmetric about the line $x=n$ (where $n$ is a constant), then the function is called an "$X(n)$ function".  
$(1)$ Among the following functions, the "$X(n)$ functions" are:  
① $y=3x-1$;  
② $y=|x|$;  
③ $y=x^{2}-x$;  
④ $y=-\frac{5}{x}$.  
$(2)$ If the function with respect to $x$, $y=(x-h)^{2}+k$, is an "$X(0)$ function", and its graph intersects the line $y=4$ at points $A$ and $B$, with the vertex of the function graph being $P$, and $\angle PBA=60^{\circ}$, find the value of $k$.  
$(3)$ If the "$X(n)$ function" with respect to $x$, $y=ax^{2}+bx-4$, passes through the point $(-1,2)$, and $n=1$, and when $t\leqslant x\leqslant t+4$, the function has a minimum value of $-6$ and a maximum value of $12$, find the value of $t$.

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