
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 triangle \( \triangle ABC \) with vertices at \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \), find the coordinates of point \( P \) on the plane such that the sum of the squares of the distances from \( P \) to the three vertices is minimized.

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