
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 with center C that passes through points A (-1, 1) and B (-2, -2), and the center of the circle lies on the line $l: x+y-1=0$.
(1) Find the standard equation of the circle with center C;
(2) If the chord intercepted by the line $kx-y+5=0$ on circle C has a length of 8, find the value of $k$;
(3) Let point P be on circle C, and point Q be on the line $l: x-y+5=0$, find the minimum value of $|PQ|$.

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