
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 a linear function $y=3x-1$ whose graph passes through points $A(a, b)$ and $B(a+1, b+k)$,

(1) Find the value of $k$;  
(2) If point $A$ is on the y-axis, find the coordinates of point $B$;  
(3) Under the condition of (2), explain whether there exists a point $P$ on the x-axis such that $\triangle BOP$ is an isosceles triangle. If it exists, directly write down the coordinates of point $P$; if not, please explain why.

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