
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
In the plane rectangular coordinate system $xOy$, the line $l: y=kx+1\left(k\neq 0\right)$ intersects the lines $x=k$ and $y=-k$ at points $A$ and $B$ respectively. The line $x=k$ intersects the line $y=-k$ at point $C$. <br/>$(1)$ Find the coordinates of the intersection point of line $l$ and the $y$-axis; <br/>$(2)$ Points with both horizontal and vertical coordinates being integers are called integer points. Let the region enclosed by the line segments $AB$, $BC$, and $CA$ (excluding the boundaries) be denoted as $W$. <br/>① When $k=2$, combined with the graph of the function, find the number of integer points inside region $W$; <br/>② If there are no integer points inside region $W$, directly write down the range of values for $k$.

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