
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 27 points P and a square with side length 1 lie on the same plane, where the vertices of the square are A, B, C, and D (counterclockwise), and the distances from P to A, B, and C are u, v, and w respectively, with the condition u^2 + v^2 = w^2, calculate the maximum distance from point P to point D.

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