
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 that line l<sub>1</sub>: y = x - 1 has two points A(x<sub>1</sub>, y<sub>1</sub>) and B(x<sub>2</sub>, y<sub>2</sub>), where x<sub>1</sub> and x<sub>2</sub> are the roots of the quadratic equation x<sup>2</sup> - 6x + 1 = 0, find the equation of the circle that passes through points A and B and is tangent to line l<sub>2</sub>: x = -1.

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