
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 the lines $x-2y=-k+6$ and $x+3y=4k+1$, if their intersection point is in the fourth quadrant.  
(1) Find the range of values for $k$;  
(2) If $k$ is a non-negative integer, and the coordinates of point $A$ are $(2,0)$, point $P$ is on the line $x-2y=-k+6$, find the coordinates of point $P$ that make $\triangle PAO$ an isosceles triangle.

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