
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
Let $P$ be a point in the plane such that the square $ABCD$ with side length $2$ also contains points $A, B, C$, and $D$, taken counterclockwise. Given the distances from $P$ to $A, B$, and $C$ are $u, v$, and $w$ respectively, where $u^2 + v^2 = w^2 + 1$, determine the maximum distance of $P$ from point $D$.

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