
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 point \( A(4,1) \), point \( B(0,4) \), and the line \( l: 3x - y - 1 = 0 \), find a point \( P \) on \( l \) such that \( |PA| - |PB| \) is maximized. Determine the coordinates of point \( P \).

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