
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 square in the coordinate plane with vertices at $A(0,0)$, $B(2,0)$, $C(2,2)$, and $D(0,2)$. Let point $P$ have coordinates $(x, y)$ in the same plane, and the distances from $P$ to $A$, $B$, and $C$ be $u$, $v$, and $w$ respectively. Under the condition that $u^2 + 2v^2 = 3w^2$, find the maximum distance from $P$ to $D$.

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