
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
In the plane, there are two points A (-1, 0) and B (1, 0). A point P is taken on the circle C: $(x-3)^2+(y-4)^2=4$.

(Ⅰ) Given $x-y+c \geq 0$ always holds, find the range of $c$.

(Ⅱ) Tangents are drawn from a point on the line $x+y+1=0$ to the circle. Find the minimum length of the tangent.

(Ⅲ) Find the extremum of $|PA|^2+|PB|^2$ and the coordinates of point P at this time.

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