
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
In triangle \( \triangle ABC \), the vertices are \((x, y, y)\) for \( i=1,2,3 \), and \( P \) is any point in the plane of \( \triangle ABC \). If the sum of the squared distances from point \( P \) to the three vertices is minimized, find the coordinates of point \( P \).

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