
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 that a point P lies in the same plane as a given square of side 2, let the vertices of the square, taken counterclockwise, be A, B, C, and D. Also, let the distances from P to A, B, and C, respectively, be u, v, and w. Assuming $u^2 + v^2 + w^2 = t^2$, find the largest possible value of PD, where t represents the distance from P to D.

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