
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 a right-angle coordinate system, given the lines \(x + y = 7\) and \(x - 7y = -33\), and the circle \(x^{2}+y^{2}-28 x+6 y+165=0\),

Determine a circle whose center is 5 times farther from the first line than from the second line, and which also passes through the intersection point of the two lines and intersects the given circle at a right angle.

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