
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
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 $D$. Define $A$ at the origin $(0,0)$, $B$ at $(2,0)$, $C$ at $(2,2)$, and $D$ at $(0,2)$. Let $u$, $v$, and $w$ be the distances from $P$ to $A$, $B$, and $C$, respectively. Find the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = 2w^2$.

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