
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 line $l: x-2y+8=0$ and two points $A(-2,8)$ and $B(-2,-4)$. If there exists a point $P$ on line $l$ such that $||PA|-|PB||$ is maximized, find the coordinates of point $P$ and the maximum value of $||PA|-|PB||$.

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