
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 circle C whose center is on the ray $3x-y=0$ ($x\geq0$), tangent to the line $x=4$, and the chord intercepted by the line $3x+4y+10=0$ has a length of $4\sqrt{3}$.
(Ⅰ) Find the equation of circle C;
(Ⅱ) For points A(1,1), B(-2,0), and point P moving on circle C, find the maximum value of $|PA|^2+|PB|^2$.

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