
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 $xOy$, there exists a circle $C$ passing through points $A(0,1)$, $B(3,4)$, and $C(6,1)$.
(Ⅰ) Find the equation of circle $C$.
(Ⅱ) If circle $C$ intersects line $x-y+a=0$ at points $A$ and $B$, and $OA \perpendicular OB$, find the value of $a$.

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