
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
Consider a point $P$ lying in the plane of a rhombus with side lengths $2$ and diagonals intersecting at right angles. The vertices of the rhombus, taken in a clockwise direction, are $A(0,0)$, $B(2,0)$, $C(3,1)$, and $D(1,1)$. If the distances from $P$ to $A$, $B$, and $C$ are $u$, $v$, and $w$, respectively, and they satisfy the condition $u^2 + w^2 = 2v^2$, determine the maximum possible distance from $P$ to $D$.

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