
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 Cartesian coordinate system $xy$, the curve $y = x^2 - 6x + 1$ intersects the coordinate axes at points that all lie on a circle $C$.
(1) Find the equation of circle $C$;
(2) If circle $C$ intersects the line $x - y + a = 0$ at two points $A$ and $B$, and $OA \perp OB$, find the value of $a$.

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