
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 $|\overrightarrow{PA}| \cdot |\overrightarrow{AB}| = \overrightarrow{PB} \cdot \overrightarrow{AB}$,  
(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 $\overrightarrow{OM} \perp \overrightarrow{ON}$, find $m$.

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