
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 the circle $C: (x-3)^2+(y-4)^2=1$, and points $A(0, -1)$ and $B(0, 1)$. Let $P$ be a moving point on circle $C$. Find the coordinates of point $P$ when the sum of the squares of distances to $A$ and $B$, $|PA|^2+|PB|^2$, reaches its maximum.

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