
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 a line segment AB with a length of $\sqrt{2}+1$, where the endpoints A and B slide along the x-axis and y-axis respectively, and a point P on AB such that $\overrightarrow{AP} = \frac{\sqrt{2}}{2}\overrightarrow{PB}$, find the equation of the trajectory of point P.

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