
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 passes through points A(1, 1) and B(2, -2), and the center C is on the line $l: x - y + 1 = 0$  
(1) Find the standard equation of the circle with center C;  
(2) If a line $l_1$ passing through point (1, 1) cuts the circle C with a chord length of 6, find the equation of the line $l_1$.

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