
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 square has vertices $A(0, 0)$, $B(1, 0)$, $C(1, 1)$, and $D(0, 1)$, and a point $P$ lies in the same plane as the square. Let $u, v, w$ be the distances of $P$ from $A, B$, and $D$ respectively. Find the maximum distance that $P$ can be from $C$ if $u^2 + w^2 = 2v^2$.

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