
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
Fold a piece of paper along a straight line $l$ once, and point $A(0,4)$ overlaps with point $B(8,0)$, point $C(6,8)$ overlaps with point $D(m,n)$.  
(1) Find the equation of line $l$;  
(2) Find the value of $m+n$;  
(3) Determine whether there exists a point $P$ on line $l$ such that $||PB|-|PC||$ has a maximum value. If it exists, find the maximum value and the coordinates of point $P$; if it does not exist, explain why.

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