
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 points $A(1,0)$, $B(-1,0)$, and a moving point $P$ on the plane satisfying $|\vec{PA}|\cdot|\vec{AB}|= \vec{PB}\cdot \vec{AB}$,  
$(1)$ Find the equation of the trajectory of point $P$;  
$(2)$ If the line $y=x+m(m\neq 0)$ intersects the trajectory of point $P$ at points $M$ and $N$, and $\vec{OM} \perp \vec{ON}$, find $m$.

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