
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
Define: If $P(x,y)$ is an integer solution of the two-variable linear equation $ax+by=c$, then point $P$ is called an "ideal point" of the two-variable linear equation $ax+by=c$.  
$(1)$ Given $A(-1,2)$, $B(4,-3)$, $C(-3,4)$, determine whether these three points are "ideal points" of the equation $2x+3y=6$, and explain the reasons;  
$(2)$ Given $m$, $n$ are non-negative integers, and $2|n|-\sqrt{m}=1$, if $P(\sqrt{m},|n|)$ is an "ideal point" of the equation $2x+y=8$, find the square root of $2m-n$;  
$(3)$ Given $k$ is a positive integer, and $p(x,y)$ is an "ideal point" of the equations $2x+y=1$ and $kx+2y=5$, find the coordinates of $P$.

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